Measures of Nonclassical Correlations and Quantum-Enhanced Interferometry

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1 University of New Mexico UNM Digital Repository Physics & Astronomy ETDs Electronic Theses and Dissertations Measures of Nonclassical Correlations and Quantum-Enhanced Interferometry Matthias Dominik Lang Follow this and additional works at: Recommended Citation Lang, Matthias Dominik. "Measures of Nonclassical Correlations and Quantum-Enhanced Interferometry." (2015). This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Physics & Astronomy ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact

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3 Measures of Nonclassical Correlations and Quantum-Enhanced Interferometry by Matthias D. Lang M.S., Physics, University New Mexico, 2011 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Physics The University of New Mexico Albuquerque, New Mexico July 2015

4 iii c 2015, Matthias D. Lang

5 iv Dedication To Kendra. Your relentless nagging finally got me to write this. And to my parents, for supporting me in most of my endeavors.

6 Acknowledgments v I want to thank all the people that in some way or another helped me in getting this dissertation done. First and foremost I am indebted to Carl Caves for the guidance, advice and financial support he provided. He was always patient, letting me work at my own pace and gave me all the freedom any grad student could dream of. His integrity in research is an inspiring example to all of his students. Furthermore I have to thank Shashank Pandey and Zhang Jiang for all the help with various aspects of continuous variable quantum mechanics when I got started on my second project and Joshua Combes for answering all kinds of questions. Thanks to Alex Tacla for letting me listen to lengthy (sometimes even interesting) monologues, Jonas Anderson for showing me how to climb, Chris Ferrie for losing at darts all the time, Chris Cesare for brewing beer with me and Julian Antolin for drinking beer with me, Rob Cook, Ben Baragiola, Leigh Norris, Jonathan Gross, Ninnat Dangniam, Ivan Deutsch, Akimasa Miyake and the rest of the hospitable CQuIC- and PandA-crowd. Outside of the physics department I need to thank my fiancée Kendra for putting up with me on a daily basis and my parents for making it possible for me to study physics in the first place. Moreover I am grateful for support in various forms from the rest of the Lang and the Lesser family. More directly related to this dissertation, I have to thank Anil Shaji, my collaborator for the work presented in Chap. 4, as well as M. Piani and S. Vinjanampathy for useful discussions pertaining to the work of Chap. 4, which was supported in part by US National Science Foundation Grant Nos. PHY and PHY The work of Chap. 5 was supported by NSF Grant Nos. PHY and PHY For the work presented in Chap. 7 I would like to thank S. Szigeti and S. Haine for useful discussions and acknowledge support from NSF Grant Nos. PHY , PHY and PHY as well as Office of Naval Research Grant No. N

7 Measures of Nonclassical Correlations and Quantum-Enhanced Interferometry vi by Matthias D. Lang M.S., Physics, University New Mexico, 2011 Ph.D., Physics, University of New Mexico, 2015 Abstract In the first part of this dissertation a framework for categorizing entropic measures of nonclassical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of nonclassical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit quantum states are presented. To demonstrate how these measures differ from entanglement we move to the set of Bell-diagonal states for two qubits, which can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and of the correlation measures from our framework for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that all of the proposed correlation measures are neither convex nor concave. In the second part we look at two practical interferometric setups that use nonclassical states of light to enhance their performance. First we consider an

8 vii interferometer powered by laser light (a coherent state) into one input port and ask the following question: what is the best state to inject into the second input port, given a constraint on the mean number of photons this state can carry, in order to optimize the interferometer s phase sensitivity? This question is the practical question for high-sensitivity interferometry. We answer the question by considering the quantum Cramér-Rao bound for such a setup. The answer is squeezed vacuum. Then we analyze the ultimate bounds on the phase sensitivity of an interferometer, given the constraint that the state input to the interferometer s initial 50:50 beam splitter B is a product state of the two input modes. Requiring a product state is a natural restriction: if one were allowed to input an arbitrary, entangled two-mode state Ξ to the beam splitter, one could generally just as easily input the state B Ξ directly into the two modes after the beam splitter, thus rendering the beam splitter unnecessary. We find optimal states for a fixed photon number and for a fixed mean photon number. Our results indicate that entanglement is not a crucial resource for quantum-enhanced interferometry. Initially the analysis for both of these setups is performed for the idealized case of a lossless interferometer. Then the analysis is extended to the more realistic scenario where the interferometer suffers from photon losses.

9 viii Contents Contents xi List of Figures xiii 1 Introduction Motivation Nonclassical correlations Interferometry with nonclassical states of light Organization of this dissertation List of publications Information Theory in the classical and the quantum world Introduction Bits Entropy Correlation and Mutual information Qubits

10 Contents ix Quantum measurements Von Neumann entropy Quantum entanglement Decoherence Maxwell s demon and Landauer s principle Discord and other nonclassical correlations beyond entanglement Introduction Discord Other forms of nonclassical correlations The set of classical states A framework for entropic measures of nonclassical correlations Introduction Framework for entropic measures of nonclassical correlations Entropic measures of information and correlation Local measurements Measures of nonclassical correlations Rank-one POVMs and projective measurements Numerical results for two-qubit states Quantum Discord and the Geometry of Bell-Diagonal States 64 6 Parameter estimation 74

11 Contents x 6.1 Classical parameter estimation and the Cramér-Rao bound Parameter estimation in a quantum setting Coherent and squeezed states Quantum noise and the shot-noise limit Nonclassical states relevant for interferometry Squeezed state interferometry NOON states Beating the Heisenberg limit Quantum-enhanced interferometry Quantum-enhanced interferometry with laser light Quantum-enhanced interferometry Fixed photon number Fixed mean photon number Quantum-enhanced interferometry with losses Lossy interferometry with laser light Lossy interferometry Conclusion 123 A Supplemental material to the correlation-measures chapters 127 A.1 The POVM inequality A.2 Nonnegativity and ordering of the WPM measure and discord

12 Contents xi A.3 Projective measurements vs. POVMs for WPM and discord A.4 Demon-based measures and rank-one POVMs B Supplemental material to the interferometry chapters 138 B.1 Lossless Interferometry (Supplemental material to chapter 7) B.1.1 Classical Fisher Information for an interferometric configuration B.1.2 Modal entanglement after the initial beam splitter B.2 Lossy interferometry (Supplemental material to Chap. 8) B.2.1 Escher et al. s bound B.2.2 Exact Fisher Information for laser interferometry with losses B.2.3 Position of the auxiliary beam splitters References 151

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14 List of Figures xiii List of Figures 2.1 (Color online) The (red) circle on the left denotes the entropy associated with system A; the (blue) circle on the right denotes the entropy associated with system B. The area on the right filled in with (blue) dots is the information missing about B given complete information about A; this area denotes the conditional entropy H(B A). Similarly, the area on the left filled in with the (red) grid denotes H(A B). The overlap between the two circles, filled with horizontal lines, denotes the mutual information H(A : B), which is the information contained in A about B and vice versa. The combined envelope of the two circles is the joint entropy H(A, B). From the diagram, we have H(B A) = H(A, B) H(A) = H(B) H(A : B) and H(A B) = H(A, B) H(B) = H(A) H(A : B). For a classical joint probability distribution, the entropic measures are all Shannon entropies or relative Shannon entropies thus they are guaranteed to be nonnegative and they are related as the diagram depicts. For a bipartite quantum state, the joint quantum von Neumann entropy, S(A, B), and the marginal von Neumann entropies, S(A) and S(B), replace H(A, B), H(A), and H(B). The measures are related as depicted in the diagram, because the quantum conditional entropies, S(B A) and S(A B), and the quantum mutual information, S(A : B), are defined by these relations. The difference is that S(B A) and S(A B), as so defined, can be negative, and thus the quantum mutual information S(A : B) can be bigger than the marginal entropies, S(A) and S(B), and bigger than the joint entropy S(A, B)

15 List of Figures xiv 4.1 M (discord) 2c = D(A B) plotted against M (WPM) 1b for one million randomly generated joint density matrices, using orthogonal projectors for the measurements. As expected, the WPM measure is never 4.2 M (dd) 3c smaller than the discord; also evident is that discord is zero for a larger class of states than the WPM measure, those being the states that are diagonal in a conditional product basis pointing from A to B. 59 plotted against M (WPM) 1b density matrices. Since M (dd) 3c for 100,000 randomly generated joint M (discord) 2c, the points from Fig. 4.1 move upwards. Many points pass the diagonal, and the ordering of Fig. 4.1 disappears M (dd) 3c plotted against M 2b for 100,000 randomly generated joint density matrices. Relative to Fig. 4.2, the points move right, due to the relation M 2b M 1b. Since not all of them pass the diagonal, there is no ordering relation between M 3c and M 2b (A) Discord (blue circles) and the WPM measure (yellow crosses) for one million randomly chosen two-qubit states, plotted against entanglement of formation, E f. As the correlations increase, the spread between entanglement and WPM or discord decreases. (B) Two superimposed histograms showing the distribution of discord and the WPM measure for ranges of values of E f : left histogram shows discord (red) and WPM (yellow) for the states of (A) corresponding to 0.1 E f 0.2; right histogram shows discord (blue) and WPM (green) corresponding to 0.3 E f

16 List of Figures xv 4.5 Deviation of the numerically obtained, optimal measurement vectors from the maximal singular vectors of the correlation matrix for the WPM measure. The joint states are a mixture of a pure product state with marginal spin (Bloch) vectors a = (1, 0, 0) and b = (1/ 2, 1/2, 1/2) and a mixed Bell-diagonal (zero marginal spin vectors) state, with correlation matrix c = diag( 0.9, 0.8, 0.7). The mixing parameter is ɛ, with ɛ = 0 corresponding to the product state and ɛ = 1 to the Bell-diagonal state. The green curve shows the cosine of the angle between the maximal right singular vector and the measurement vector on system B. The red curve is the cosine of the angle between maximal left singular vector and the measurement vector on system A Geometry of Bell-diagonal states. The tetrahedron T is the set of valid Bell-diagonal states. The Bell states β ab sit at the four vertices, the extreme points of T. The green octahedron O, specified by c 1 + c 2 + c 3 1 (λ ab 1/2), is the set of separable Bell-diagonal states. There are four entangled regions outside O, one for each vertex of T, in each of which the biggest eigenvalue λ ab is the one associated with the Bell state at the vertex. Classical states, i.e., those diagonal in a product basis, lie on the Cartesian axes Level surfaces of the quantum mutual information I (left column) and the accessible classical information C (right column): (a) I = 0.1; (b) I = 0.55; (c) C = 0.15; (d) C = 0.4. The smooth surface of cali = 0.1 bulges towards the vertices of the tetrahedron T from Fig. 5.1; as the mutual information grows, this surface becomes inflated and eventually intersects T, giving rise to the windows seen in (b) for I = The level surfaces for the accessible classical information C are cubes centered at the origin. As C increases corners of the cube get cut of as they poke through the surface of T

17 List of Figures xvi 5.3 Surfaces of constant discord: (a) D = 0.03; (b) D = 0.15; (c) D = The level surfaces consist of three intersecting tubes running along the three Cartesian axes. The tubes are cut off by the state tetrahedron T at their ends, and they are squeezed and twisted so that at their ends, they align with an edge of T. As discord decreases, the tubes collapse to the Cartesian axes [DacVB10]. As discord increases, the tube structure is obscured, as in (c): the main body of each tube is cut off by T ; all that remains are the tips, which reach out toward the Bell-state vertices Trajectory (red) of a Bell-diagonal state under random phase flips of the two qubits; initial conditions are c 1 (0) = 1, c 2 (0) = c 3 (0) = 0.3. The trajectory is the straight line c 3 = c 3 (0) = 0.3 = c 2 /c 1. For clarity, only the (+,, +)-octant is shown. A constant-discord surface is plotted for the discord value of the initial state. Faces of the yellow state tetrahedron T and the green separable octahedron O are also shown. The straight-line trajectory proceeds along a tube of constant discord till it encounters the vertical tube at c 1 = 0.3, after which discord decreases monotonically to zero when the trajectory reaches the c 3 axis. Entanglement of formation decreases monotonically to zero when the trajectory enters O at c 1 = 0.7/1.3 = It is convenient to think of parameter estimation in a quantum setting as a three-step process: 1) A probe is prepared, i.e., the relevant quantum system is prepared into a known state. 2) The known probe state evolves according to a parameter-dependent process. 3) A measurement is performed and from the outcomes of this measurement the parameter of interest is estimated

18 List of Figures xvii 7.1 Measurement of a differential phase shift. An (upper) mode a in a coherent state α and a (lower) mode b in an arbitrary pure state χ are incident on a 50:50 beam splitter, which performs the unitary transformation B of Eq. (7.2). After the beam splitter, phase shifts ϕ 1 and ϕ 2 are imposed in the two arms; the action of the phase shifters is contained in the unitary operator U of Eq. (7.3). Finally, a measurement is made to detect the phase shifts. When the measurement is pushed beyond a second 50:50 beam splitter, the result is a Mach-Zehnder interferometer, which is sensitive only to the differential phase shift φ d = ϕ 1 ϕ Modified interferometric setting for the analysis including linear losses. While most of this setup is identical to Fig. 7.1, two auxiliary beamsplitters, shown in blue, both with transmissivity η, are introduced to model linear losses in the two arms, here assumed to be identical in the two arms. The phase shifts φ 1 and φ 2 in the ancillary modes after the interaction with the main system do not change any physical quantity we are considering; they should be thought of as a mathematical trick to obtain a better bound for the quantum Fisher information B.1 First half of our interferometric setup. The ancillary modes start out in the vacuum state. The 50:50 beam splitter of the interferometer B is drawn in black, while the ancillary beam splitters, B 1 and B 2, with transmissivity cos 2 (µ/2), are depicted in blue B.2 (a) This physical situation is equivalent to the one pictured in Fig. (B.1). (b) The first beam splitter between modes c and d has been removed as it does not change anything, given that both ancillary modes start out in vacuum

19 1 Chapter 1 Introduction 1.1 Motivation With the rapid technological development of the last century, the performance of nearly any device one can think of has been dramatically improved. The generic example here is the exponential increase in transistor densities of integrated circuits, known as Moore s law. While this predicted growth has been observed for almost fifty years, the first signs of its slowdown might soon become apparent [ZHB03]. This is not really surprising given the discrete structure of matter: A transistor cannot be smaller than one atom. One can still increase the computing power at this point, however, by simply using more resources and increasing the size of the computer. Quite generally, limits on how well any device can perform, given a defined set of resources, are imposed by Nature. As Nature is described using quantum mechanics, the most fundamental limitations on device performance need a quantum-mechanical treatment. While the description needs to be quantum mechanical to deliver accurate predictions, an interesting distinction can be made between devices using resources of a classical nature and devices using a more general class of resources that do not have a classical description. The goal of research in quantum information theory (QIT), for

20 Chapter 1. Introduction 2 example, is to classify and pin down the differences between information-processing protocols that use quantum resources and those protocols that have a full description in classical information theory. Arguably, most of the researchers in quantum information theory are drawn to this field because of the exciting prospect it offers to tackle information-processing tasks that have been deemed infeasible on classical information processors no matter what their size is. The famous example that every student of the field cites when justifying their research to people not familiar with QIT, is Shor s algorithm. This algorithm allows factoring of integers on a quantum computer in polynomial runtime, a problem on whose classical hardness the security of the widely used RSA cryptosystem is based on. While Shor s algorithm presents the most striking selling point for research in QIT, current technology is still decades away from implementing a version of Shor s algorithm that could be used by people interested in breaking RSA encryption. Moving from the ambitious goal of performing a task not feasible classically, such as Shor s algorithm or the idea of quantum simulation, to the more modest goal of simply performing a task which is feasible classically, but that can be done better using quantum resources, we can see various examples, where research in quantum information theory has led to ideas that either have been implemented or are at least implementable with current or near-future technology. Quantum cryptography systems are commercially available, and the use of quantum resources in metrological setups allowed for the design of quantum clocks, accurate to one second in 3.4 billion years and thus able to demonstrate gravitational time dilation within a single room. Nonclassical resources are also used to upgrade LIGO, the world s most sophisticated and powerful laser interferometer, in the hopes that this will push its sensitivity far enough to make the first direct observation of gravitational waves a reality. On the one hand, research in QIT is devoted to analyzing practical situations and determining how quantum resources have to be deployed in these situations to provide a benefit. On the other hand, scientists working in the field explore the foundations of the theory, trying to get a better grasp on what it is in the very structure of quantum mechanics that is responsible for these quantum enhancements.

21 Chapter 1. Introduction 3 The research presented in this dissertation touches on both of these branches. In the first part nonclassical correlations beyond entanglement are discussed. These correlations might turn out to be a key piece in the puzzle of pinning down the resource that enables quantum systems to process information more efficiently than their classical counterpart. The second part is devoted to analyzing the ultimate limits on a practical interferometric setup that is allowed the use of nonclassical states of light. Such an interferometer can be used to measure various quantities, such as small distances, more precisely than possible with a standard laser interferometer and therefore could provide new insights in various fields of research. 1.2 Nonclassical correlations In order to represent quantum information efficiently a quantum-information-processing system has to be composed of parts [BKCD02]. For multi-partite systems, correlations between these parts can arise that are conceptually different from correlations between classical systems. One kind of nonclassical correlation is entanglement [HHHH09]. Entanglement is the crucial resource for such quantum-information-processing protocols as quantum key distribution, teleportation, and super-dense coding [HHHH09]. Moreover, Josza et al. [JL03] and Vidal [Vid03] showed that entanglement is a resource for pure-state quantum computation. A pure quantum state is unentangled if it is a product of pure states for each part. Josza et al. and Vidal provided a classical algorithm that efficiently simulates any pure state quantum computation where entanglement is negligible (more precisely, where entanglement is restricted to blocks of processing elements of a fixed size). While the exact amount of entanglement required for a pure-state quantum algorithm that promises an exponential speedup over a classical one performing the same task can be quite small [VdN13], Josza et al. s and Vidal s results nevertheless prove that entanglement is necessary. The constructions by Josza et al. [JL03] and Vidal [Vid03] relies on the fact that there exists an efficient description of pure states that are not entangled.

22 Chapter 1. Introduction 4 For mixed states, the situation is more complicated. A mixed state is unentangled (separable) if it can be written as an ensemble of pure product states. Operational measures of entanglement are notoriously difficult to calculate for mixed states and even the boundary between separability and entanglement is difficult to characterize. In general, one can say, however, that the set of separable states is a convex set, is invariant under local unitary operations, and has dimension (as a manifold) as large as the space of mixed states [HHHH09]. This dimension, unfortunately, is the reason why for unentangled mixed states, no efficient description is available and the construction from [JL03] cannot be applied. It is unknown whether entanglement is necessary for an exponential speedup in an arbitrary and thus mixed-state quantum computation. In the realm of mixed states, separable states can have nonclassical correlations even though they are unentangled. A state with only classical correlations, often called a classical state, is one that is diagonal in a product basis, for then the correlations are described by a joint probability distribution for classical variables of the parts. These purely classical states are a set of measure zero; this is suggested by the fact that any classical state can be perturbed infinitesimally to become nonclassical by making two of the eigenvectors infinitesimally entangled and is proved rigorously in [FAC + 10]. The fact that the states without any nonclassical correlations are a set of measure zero begs the question of whether there is an efficient description of these kinds of states. If so, a similar construction to the one proposed by Josza and Vidal might be used to show that a more general type of nonclassical correlation, not just entanglement, is a requirement for exponential speedups in quantum algorithms. Eastin [Eas10] followed this agenda and investigated whether a computation only involving classical states can be simulated efficiently on a classical computer. While he was able to show that computations comprised of qubits and gates involving not more than two qubits, provided the state remains a product state after all gates, can indeed be simulated efficiently on a classical computer, his result is apparently not extendable to a more general setting, and no one has taken up the challenge of finding

23 Chapter 1. Introduction 5 a more general result in this direction. Another result hinting at a possible connection between nonclassical correlations beyond entanglement and quantum speedups is related to the model deterministic quantum computation with one quantum bit (DQC1) [KL98]. While this model is not universal, it can perform at least one specific task for which there is no efficient classical algorithm known. It was shown in [DFC05] that in this model only a negligible amount of entanglement is present; Datta et al. [DSC08], however, demonstrated the presence of another type of nonclassical correlation, called quantum discord, in typical instances of DQC1. While the connection of nonclassical correlations to quantum speedups remains tenuous [DacVB10], even this tenuous connection sparked a considerable interest. A variety of measures have been proposed to quantify nonclassical correlations for bipartite systems [OZ01, OHHH02, Luo08a, PCMH09, WPM09, BT10, LCS11, MPS + 10], in ways that can be nonzero for separable, but nonclassical states. Our contribution to this field is to present a framework that unifies several of the proposed measures of quantum correlations that go beyond entanglement. While some new measures emerge from this framework, its main purpose is to investigate how the proposed measures relate to one another and to determine whether there is anything special about several existing measures, including quantum discord. Moving to Bell-diagonal states, a specific set of two-qubit states parametrized by three real parameters, allows us then to illustrate some of the properties of these correlation measures; conveniently all the different measures of non classical correlations agree on this subset of two-qubit states, and we can use a pictorial approach to explain some of the properties that are common to all these measures.

24 Chapter 1. Introduction Interferometry with nonclassical states of light The second topic that will be discussed within this thesis moves away from the foundational topic of quantum correlations to a more practically oriented problem in the field of quantum metrology. Specifically we will investigate how to use the quantum resource of nonclassical light fields most efficiently to optimize the performance of a practical interferometer. The discovery that squeezed vacuum, injected into the normally unused port of an interferometer, provides phase sensitivity below the shot-noise limit [Cav81] led to thirty years of technology development, beginning with initial proof-of-principle experiments [XWK87, GSYL87] and culminating recently in the use of squeezed light to beat the shot-noise limit in the GEO 600 gravitational-wave detector [C + 11] and the Hanford LIGO detector [AAA + 13]. In the last decade much work has been devoted to exploring ultimate quantum limits on estimating the differential phase shift between two optical paths and to finding the states that achieve these limits. Given exactly N photons, the optimal state, in the absence of photon loss, is a N00N state, ( N, 0 + 0, N )/ 2 [Ger00, BKA + 00, GBC02, LKD02], i.e., a superposition of all photons proceeding down one path with all photons proceeding down the other path. The N00N state is the optical analogue of the cat state that is optimal for atomic (Ramsey) interferometry [BIWH96]. Ideally, the sensitivity of a phase estimation setup using N00N states scales linearly with the energy used to produce the state. This provides a quadratic improvement over the best scaling possible when only employing classical resources, where the particle nature of photons gives rise to the shot-noise limit. Since the N00N state is extremely sensitive to photon loss, considerable effort has gone into determining optimal N-photon input states and corresponding sensitivities in the presence of photon loss [DDDS + 09, KSD11, EdMFD11a, EdMFD11b]. While these states indeed deliver optimal or near-optimal performance, given a

25 Chapter 1. Introduction 7 fixed input energy, we argue that they are not of practical relevance because they are very hard to produce with current technology and are therefore only available with quite low photon numbers. Consequently, the phase resolution obtained from using these optimal states cannot compete, even remotely, with the resolution obtained from a classical interferometer operating at or near the shot-noise limit with a strong, commercially available laser. This does not mean, however, that nonclassical states are useless for metrology. The use of squeezed states to enhance the sensitivity of the GEO 600 and LIGO interferometers is testimony to the efficacy of squeezed light in a situation where the lasers powering the interferometer have been made as powerful as design constraints allow. In the work we present here, we turn the focus away from interferometry with states which can only be created with very small numbers of photons and instead start from the assumption of a more practical interferometric setting. We will still search for nonclassical input states that perform best, but we will do so under additional constraints that ensure that the states we find are not just of academic interest. Inspired by Caves original proposal, we will look at an interferometer whose main source of power is a laser and ask the question: What is the optimal nonclassical state to inject into the second input port to maximize the interferometers phase sensitivity? In this way the main power production is separated from the production of nonclassical light. One can still make use of strong commercially available lasers, while the exotic state of light does not need to be powerful to obtain a significant improvement of the devices performance. We will prove that squeezed vacuum is the optimal choice in this setting, thus showing that in this sense the current LIGO setup is optimal. The main problem in the production of N00N states is the challenge to entangle two optical modes. To avoid this problem, another sensible question of practical importance is the following: what is the optimal unentangled input state to an interferometer? If we pursue this approach, any entanglement necessary for enhanced resolution gets produced by the first beam splitter of the interferometer. We analyze this situation for the two cases of a fixed photon number and mean photon number

26 Chapter 1. Introduction 8 constraint. Making this simple assumption, that the state in the two input modes of the interferometer is of product form, we find for the case of fixed photon number, that the optimal sensitivity is given by identical or fraternal twin-fock states, depending on whether the photon number is even or odd. For the case of a mean-photon-number constraint, we will see that squeezed vacuum, now in both input modes, again yields the best sensitivity. While this product state of two squeezed vacua has essentially the same sensitivity as a N00N state, it happens to be an eigenstate of the first beam splitter of the interferometer. As a consequence, no modal entanglement is generated at all before the state is subjected to the phase shift. Hence, contrary to what one might expect given the discussion of N00N states, our findings indicate that modal entanglement is not a crucial resource in quantum-enhanced interferometry. The analysis of the two scenarios described above is first performed under the assumption of an idealized interferometer, that is an interferometer which is not subjected to photon losses. We make use quantum Fisher information [Hel76, Hol11, BC94] as the figure of merit to quantify the performance of a particular input state to determine the phase shift occurring in the interferometer. The main source of noise that any linear optical setup is subjected to is loss of photons. Consequently any real world interferometer has to deal with photon losses. In recent years it has been shown by several groups [FI08, DDDS + 09, KSD11, EdMFD11a] that losses can be very detrimental to the performance of any state in an interferometric setup. Most important was the observation that asymptotically, as the mean photon number employed to produce a state goes to infinity, no state is able to achieve the sought after Heisenberg scaling, the quadratic improvement over the best classical scaling displayed by N00N states in the lossless regime. In the light of this, an analysis that includes photon losses is desirable whenever ultimate limits of sensitivities are discussed. Our case, being chiefly motivated by practical concerns, makes a close account of the effects of losses mandatory. In the last part of this dissertation, we revisit the analysis of the optimal state for a laser-powered interferometer and for an interferometer powered by an arbitrary product state, given a mean photon number constraint, and extend the analysis to the case where photon

27 Chapter 1. Introduction 9 losses occur in the setup. Losses transform pure states to mixed states and unlike for the pure-state case, there is no general, explicit expression for the quantum Fisher information available. This makes an analysis including losses much harder, as it is not possible to incorporate losses into calculations analogous to those performed for the lossless case. Using some additional tools, however, we are able to show that for the cases of practical interest, the optimal states for the lossless case remain optimal (or very nearly so) even when photon losses degrade the performance of the interferometer. 1.4 Organization of this dissertation As stated above, this dissertation is comprised of contributions to two topics in quantum information theory, so it is divided into two distinct parts. The first part, Chaps. 2 5, deals with the foundational topic of nonclassical correlations that go beyond the notion of quantum entanglement, while the second part, Chaps. 6 8, treats a more practical question in the field of interferometry. Within the first part, Chap. 2 presents some concepts of classical information theory and their counterparts in quantum information theory. It provides the background and introduces notation used in the first part of the dissertation. Chapter 3 discusses some results on measures of nonclassical correlations, some of which are investigated in more detail in Chap. 4, the central chapter for this first part. In this central chapter, a framework is introduced that demonstrates the relations between various measures of quantum correlations that have been proposed independently and that highlights the special properties of particular measures including quantum discord. The first part concludes with Chap. 5, where we investigate specific properties of these nonclassical correlation measures on a convenient subset of two-qubit states. The second part of the dissertation starts in Chap. 6 with a basic discussion of parameter estimation and, more specifically, why Fisher information can be used as a figure of merit when one is interested in the sensitivity of a particular estimation

28 Chapter 1. Introduction 10 scheme. Chapters 7 and 8 give the central results of the analysis. Chapter 7 investigates optimal input states to an interferometer under certain constraints motivated by practical considerations in the absence of losses. This relatively simple, but important analysis is extended in Chap. 8 to the situation where the interferometer is subject to photon losses. The analysis that includes losses requires considerably more technical apparatus than the lossless analysis, but confirms its main conclusions about which input states are optimal. 1.5 List of publications The following provides a list of publications encompassing most of the work presented in this dissertation. 1. M. D. Lang and C. M. Caves, Quantum discord and the geometry of Bell-diagonal states, Physical Review Letters 105, (2010). 2. M. D. Lang, C. M. Caves and A. Shaji, Entropic measures of non-classical correlations, International Journal of Quantum Information 9, (2011). 3. M. D. Lang and C. M. Caves, Optimal quantum-enhanced interferometry using a laser power source, Physical Review Letters 111, (2013). 4. Z. Jiang, M. D. Lang and C. M. Caves, Mixing nonclassical pure states in a linear-optical network almost always generates modal entanglement, Physical Review A 88, (2013). 5. M. D. Lang and C. M. Caves, Optimal quantum-enhanced interferometry, Physical Review A 90, (2014). 6. S. A. Haine, S. S. Szigeti, M. D. Lang, and C. M. Caves, Heisenberg-limited metrology with information recycling, Physical Review A, 91, (R) (2015). The first two items in the list report the results of the first part of the dissertation, i.e., the results on measures of nonclassical correlations. Items 3 6 are related to the

29 Chapter 1. Introduction 11 second part of the dissertation, i.e., the results on practical interferometry. Items 3 and 5 are concerned with lossless interferometry in the two scenarios discussed in Chap. 7. Item 4 presents related results on how entanglement is generated in linear optical networks, and item 6 discusses applications of quantum Fisher information to an interferometric technique called information recycling. The results in items 4 and 6 are not discussed in the dissertation. The work on lossy interferometers, reported in Chap. 8, is in preparation for publication.

30 12 Chapter 2 Information Theory in the classical and the quantum world 2.1 Introduction In this chapter I will present some basic concepts and quantities important in classical information theory and their generalization to the quantum world. My goal here is to describe these concepts such that someone outside of the field of quantum information can follow the main points of this dissertation. I do not intend to be rigorous when introducing some mathematical concepts, but will instead try to provide simple examples to encourage an intuitive understanding of these concepts. A detailed discussion of most of the following can be found in [NC00]. 2.2 Bits Arguably the most fundamental concept in all of information theory is the concept of a binary digit or as it is commonly called a bit. A bit is some system that can be in either of two states. Bits are used to store information. Physically, say in a computer, this is typically done on an integrated circuit, where voltage in a particular part of the

31 Chapter 2. Information Theory in the classical and the quantum world 13 circuit is used to represent a bit. The two states a bit can take are represented by a high and a low voltage. In the following we will not be concerned about the physical realization of a bit, but rather talk about the abstract concept. This abstraction away from particular physical realizations is the hallmark of an information theory. As is standard practice, I will label the two states a bit can take by 0 and Entropy The most basic quantity in classical information theory, used to quantify information, e.g., the information carried by a bit, is the Shannon entropy H. Given a random variable A with an associated probability distribution (or probability mass function) p a, the Shannon entropy H(A) of this random variable is given by [NC00] H(A) a p a log p a. (2.1) We will use the convention here and in the following that logarithms are to be taken to base 2. This will fix the unit of information to be bits. The above quantity tells us the amount of information we gain, on average, from learning the value of the random variable A. An example of this would be a coin toss. Having two possible outcomes, heads and tails, a coin toss can be thought of as a random bit A. Even though I will be talking about a coin toss, I will use the standard convention for bits 0 and 1 to label the outcomes. For a fair coin toss, we have p 0 = p 1 = 1 2 H(A) = 1. (2.2) This means we gain 1 bit of information by looking at the outcome of the fair toss. On the other hand, if we look at the outcome of a maximally biased coin, with p 0 = 0, p 1 = 1 H(A) = 0, (2.3) we will gain no information, as there was no uncertainty to start with: the outcome of the toss was determined before we tossed the coin.

32 Chapter 2. Information Theory in the classical and the quantum world 14 A strong operational motivation of why it is expression (2.1) that is commonly used to quantify the information is provided by Shannon s source coding theorem [CT12], which links H(A) to the minimal average length of bit strings that faithfully represent the information in the random source A. If we consider one additional system, say another bit B, we will use the joint entropy H(A, B) to quantify the overall information in this joint system. By analogy with the above, the joint entropy is defined as H(A, B) a,b p a,b log p a,b, (2.4) where p a,b is the joint probability density (mass) function, for system A being in the state a and system B being in the state b. Again take a coin toss as an example. We can imagine two independent and fair coins tossed together. We have p 0,0 = 1 4 = p 0,1 = p 1,0 = p 1,1 H(A, B) = 2. (2.5) It makes sense intuitively that the information gained from 2 independent fair coins is twice the amount from a single fair coin. In the next section we will see what happens when the two coins are not independent Correlation and Mutual information The two random variables A and B can be correlated, in which case they share some information. As an example for perfect correlation, one could image two fair coins glued together by their edges, both facing heads up. The outcome of coin A is still completely random, but once we look at coin A, we know the outcome of coin B. Looking again at the probabilities and entropies, we see p 0,0 = 1 2 = p 1,1, p 0,1 = 0 = p 1,0 H(A) = 1 = H(B) = H(A, B). (2.6) Here we see that the total information we can gain is present in either coin; looking at the other reveals nothing new. We can write this as H(A B) = 0 = H(B A), (2.7)

33 Chapter 2. Information Theory in the classical and the quantum world 15 where H(A B) is to be read as the entropy of A given that that we know the state of B, H(A B) H(A, B) H(B). (2.8) In this last example, one coin shares all its information with the other. A quantifier for this shared information is known as the mutual information, H(A : B) H(A) + H(B) H(A, B). (2.9) For two perfectly correlated coins we have H(A : B) = H(A, B), which restates the fact that all of the information in this bipartite system is shared information. As should be obvious from the motivation, each of the quantities introduced above is a positive quantity. This enables us to display how these quantities relate in the Venn diagram of Fig Qubits A quantum bit, or a qubit for short, is the quantum version of a classical bit and the smallest conceivable quantum system. While a classical bit can be in one of two states, a qubit s state can be represented by any (normalized) vector of a two-dimensional Hilbert space H 2. Using the standard bra-ket notation we use the symbol a to denote a vector a and a to denote its dual vector. To write down the state of a pure qubit, we typically pick an orthonormal basis. The most common one is called the computational basis and uses the basis vectors 0 and 1. We will use this basis in the following, unless we explicitly specify another one. Now we can write an arbitrary (pure) qubit state ψ as ψ = α 0 + β 1, (2.10) where α and β are complex numbers such that α 2 + β 2 = 1. It turns out that the global phase of a quantum state ψ is not observable and can be neglected, ψ e iϕ ψ, (2.11)

34 Chapter 2. Information Theory in the classical and the quantum world 16 Figure 2.1: (Color online) The (red) circle on the left denotes the entropy associated with system A; the (blue) circle on the right denotes the entropy associated with system B. The area on the right filled in with (blue) dots is the information missing about B given complete information about A; this area denotes the conditional entropy H(B A). Similarly, the area on the left filled in with the (red) grid denotes H(A B). The overlap between the two circles, filled with horizontal lines, denotes the mutual information H(A : B), which is the information contained in A about B and vice versa. The combined envelope of the two circles is the joint entropy H(A, B). From the diagram, we have H(B A) = H(A, B) H(A) = H(B) H(A : B) and H(A B) = H(A, B) H(B) = H(A) H(A : B). For a classical joint probability distribution, the entropic measures are all Shannon entropies or relative Shannon entropies thus they are guaranteed to be nonnegative and they are related as the diagram depicts. For a bipartite quantum state, the joint quantum von Neumann entropy, S(A, B), and the marginal von Neumann entropies, S(A) and S(B), replace H(A, B), H(A), and H(B). The measures are related as depicted in the diagram, because the quantum conditional entropies, S(B A) and S(A B), and the quantum mutual information, S(A : B), are defined by these relations. The difference is that S(B A) and S(A B), as so defined, can be negative, and thus the quantum mutual information S(A : B) can be bigger than the marginal entropies, S(A) and S(B), and bigger than the joint entropy S(A, B). which enables us to parametrize the qubit state in terms of two angles θ and φ, ψ = cos θ eiφ sin θ 1. (2.12) 2 Given this parametrization we can geometrically think of the state of a pure qubit as a point on a sphere, the so-called Bloch sphere. Orthogonal states in this picture

35 Chapter 2. Information Theory in the classical and the quantum world 17 lie at antipodal points on this sphere. A convention we will adopt is to identify the state 0 with the north pole of the sphere and 1 with its south pole. Any spin- 1 object, say the spin of an electron, is an example of this simplest of 2 quantum objects. More commonly, however, in the lab two isolated levels of a more complex system such as an atom are used as a qubit. The qubit we have been talking about up to now was always in a pure state. In analogy to a random classical bit, we will allow qubits to be in a statistical mixture of pure states, say an equal mixture of 0 and 1. These states are simply known as mixed states. Mixed states are convex combinations of more than one state and we will use the density matrix formalism to describe them. The density matrix ρ is a positive semi-definite matrix that describes a quantum state. The density matrix ρ of a pure state ψ is simply the outer product of is state vector with itself: ρ = ψ ψ. (2.13) To get the density matrix ρ mix for the example above, we form the convex combination of states 0 0 and 1 1 : ρ mix = p (1 p) 1 1. (2.14) Written in the computational basis this becomes: ρ mix = p 0. (2.15) 0 1 p For an equal mixture we have p = 1/2. superposition ( )/ 2, whose density matrix is ρ sup = Notice that this is different from the. (2.16) In the picture of the Bloch sphere, mixed states sit in the inner part of the sphere. The equal mixture of 0 and 1 for example is the point associated with the center of the sphere.

36 Chapter 2. Information Theory in the classical and the quantum world 18 A common way of writing the density matrix of a qubit makes use of the Pauli matrices σ = (σ x, σ y, σ z ) as a hermitian operator basis: σ x = 0 1, σ y = 0 i, σ x = 1 0. (2.17) 1 0 i With this, a qubit state ρ is specified by a vector v in the Bloch-sphere picture, ρ = 1 2 (I 2 + v σ), (2.18) where v is called the Bloch vector. The Bloch vector of the mixed qubit from Eq. (2.14) is v mix = (0, 0, 2p 1), while the superposition in Eq. (2.16) has a Bloch vector of v sup = (1, 0, 0). As these examples illustrate, the Bloch vector of a pure state is normalized, while the Bloch vector of a mixed states are sub-normalized. If we have a composite system, e.g. two qubits, the vector space H comp that the new state lives in will be the tensor product of the two Hilbert spaces of its constituents, a 4-dimensional Hilbert space for two qubits H 4 = H 2 H 2. To denote state vectors of a joint system, we will often omit the tensor product sign and write a b for a b, if it is obvious from the context what this notation will mean. Using this convention, the standard computational basis for the combined space will be { 00, 01, 10, 11 }. Using the two states ρ mix and ρ sup as an example, their joint density matrix ρ joint is: p ρ joint = ρ mix ρ sup = (1 p) (2.19) Conversely we will use the partial trace to obtain the density matrix for one of the marginal states from the density matrix of the joint system. An index on the trace specifies which system is traced out. Using the example above we have tr B (ρ joint ) = p( 1 + 1) 2 2 0( 1 + 1) 2 2 = ρ mix, (2.20) 0( 1 + 1) 2 2 (1 p)( 1 + 1) 2 2

37 Chapter 2. Information Theory in the classical and the quantum world 19 and tr A (ρ joint ) = 1(p + 1 p) 1(p + 1 p) (p + 1 p) 1(p + 1 p) 2 2 = ρ sup. (2.21) Similar to Eq. (2.18) we can write arbitrary two qubit states as ρ = 1 4 ( I 4 + a σ I 2 + I 2 b σ + i,j c i,j σ i σ j ). (2.22) Here a and b are the Bloch vectors of the respective marginal states; c i,j = σi A σj B is a matrix defined in terms of expectation values of products of Pauli operators and contains information about the correlation of the two subsystems. Again for the sake of a more compact notation, we will sometimes omit the tensor product sign. Especially when an operator O acts only on one system, say the Hilbert space of system H A of a composite system with Hilbert space H A H B, we will write O A meaning O I Quantum measurements Unlike a classical system, whose state we can measure without disturbing it, a quantum mechanical measurement is intrinsically invasive and will generally change the quantum state of the system being measured. We will use the positive-operatorvalued-measure (POVM) formalism to mathematically describe a measurement. While the POVM formalism describes more general measurements than the projective measurements usually discussed in textbooks, it only tells us about the statistics of the measurement and not about the state of the quantum system after the measurement has been performed. A full description of the measurement process requires using what are called quantum operations, which is beyond the scope of this brief introduction. This, however, should not overly concern us as these post-measurement states will not be relevant for most of the following discussion, although in one section quantum operations pop up briefly. A complete discussion of the general measurement formalism can be found in [NC00].

38 Chapter 2. Information Theory in the classical and the quantum world 20 A POVM is defined by a set of positive operators, {E m }. The members of the set are called POVM elements. The POVM elements satisfy the completeness relation E m = I. m (2.23) The probability p m for measurement outcome m, when measuring on a quantum state ρ, is given by p m = tr(e m ρ). (2.24) If the set of POVM elements consists of orthogonal projectors, the measurement is called a projective measurement. The POVM elements for a measurement in the computational basis are an example of a projective measurement with rank-one projectors: E 0 = 1 0, E 1 = 0 0. (2.25) If we measure the equal mixture of the state ( )/2 in this basis, we get equal probabilities p 0 = 1/2 = p 1. At this point we can highlight the crucial difference between mixed states and superpositions of states. Looking at the measurement statistics for the state ρ sup in Eq. (2.16), we see that it equals the statistics of the mixed state above. Changing the measurement basis to the eigenbasis of ρ sup, however, E + = , E = , (2.26) reveals that these two states differ drastically: p + = 1 2, p = for the mixed state, but p + = 1, p = 0 for ρ sup. Pure states can always be measured in their eigenbasis with certainty, but for mixed states, any rank-one POVM has at least two outcomes with nonzero probability Von Neumann entropy Mixed states are the quantum analogue of a random bit, whereas pure states are the analogue of a bit in a defined state. Given this analogy, we expect a quantum entropy

39 Chapter 2. Information Theory in the classical and the quantum world 21 measure to be zero for a pure state, but greater than zero for a mixed state. The von Neumann entropy S(A) of a quantum system A with density matrix ρ, S(A) = S(ρ) tr(ρ log ρ), (2.27) fulfills these expectations and is the quantum analogue of the Shannon entropy. As is standard convention, functions of operators are applied to the eigenvalues in the eigenbasis of the operator. It turns out that the von Neumann entropy shares many of the properties the Shannon entropy [NC00]. Similar to its classical counterpart, Schumacher s quantum noiseless channel coding theorem gives the von Neumann entropy an operational meaning by connecting it the the minimum average number of qubits needed to reliably represent the quantum information contained in ρ. Similar to the classical case we can add a second system and ask about the quantum information shared by these two systems. The generalization of joint entropy is straightforward. Mutual information can be written analogously to the classical case, S(A : B) S(A) + S(B) S(A, B). (2.28) The most obvious choice for the quantum conditional information S(A B) is S(A B) = S(A, B) S(B). (2.29) One has to be careful, however, when looking at Fig. 2.1 to visualize these quantities. Most notable is that the quantum analogue of the conditional information is no longer necessarily a positive quantity. We will see an example of this in the next section on entanglement. Moreover, interpreting the conditional information as the information about system X given that we learned what state system Y is in, is not as straightforward as in the classical case. Learning the state of a system involves a quantum measurement, which generally alters the state itself, and it seems we would have to use the quantum measurement formalism to define an appropriate conditional information. In Chap. 3, we will meet an alternative definition for the the conditional entropy based on this thought.

40 Chapter 2. Information Theory in the classical and the quantum world Quantum entanglement Entanglement is a form of correlation between two quantum systems, which has no classical counterpart. The defining characteristic of a pure entangled state is that both of the marginal states are mixed. In fact the entropy of the marginal state is used to quantify the entanglement of the overall state. The canonical example for an entangled state is a Bell state β 00 = ( )/ 2. (2.30) We have tr A (ρ β00 ) = I/2 = tr B (ρ β00 ) and S(tr A (ρ β00 )) = 1. We say this state carries one bit of entanglement. The entropy of the marginal states is an entanglement measure called the entropy of entanglement. Unfortunately this only works for pure states. If we go to the realm of mixed states, entanglement is more difficult to quantify. What we can say is that a state ρ sep is not entangled when its density matrix can be decomposed into product states: ρ sep = i p i ρ A i ρ B i. (2.31) We say such a state is separable. While this boundary between entangled and unentangled states is clearcut, there are different ways of quantifying the entanglement. Several operationally motivated quantities such as entanglement of formation, squashed entanglement, entanglement cost, and a variety of others have been proposed [HHHH09]. These operational measures limit to the entropy of entanglement for pure states, but are notoriously difficult to calculate for mixed states; even the boundary between separabiltiy and entanglement is difficult to characterize, despite its seemingly simple definition. From Eq. (2.31), however, it is obvious that the set of separable states is a convex set. Moreover, it is known that the set is invariant under local unitary operations and has dimension as large as the space of mixed states [HHHH09]. Entanglement is the crucial resource for several quantum-information-processing protocols, such as quantum key distribution, teleportation and super-dense coding [HHHH09].

41 Chapter 2. Information Theory in the classical and the quantum world 23 As we alluded to in the previous section, entangled states are an example of where the analogy to the classical entropic quantities breaks down. The conditional entropy of the Bell entangled state, for example, is negative, i.e., S(A, B) S(A) = 0 1 = 1. So in some sense the overall state is defined while its constituents are completely random. This seems counter-intuitive, which is not surprising given that there is no analogue in the classical world to this phenomenon Decoherence Decoherence is a process that reduces the purity of a state due to a system s interaction with other degrees of freedom, which one does not consider part of the system one is interested in. Commonly one calls those degrees of freedom collectively the environment. Decoherence turns pure states into mixed states and during this process may destroy quantum correlations. It is responsible for the fact that we do not encounter quantum effects in macroscopic systems, such as a superposition of a living and a dead cat. Decoherence poses a fundamental obstacle when trying to control a quantum system for a specific purpose. Quantum systems that are intrinsically well insulated from the environment and therefore robust against decoherence are generally hard to control because of their inaccessibility. On the other hand, quantum systems that are easy to control usually also couple strongly to the environment and therefore lose their quantumness easily. Mathematically, decoherence can be described by the decay of off-diagonal elements of the density matrix in some basis preferred by the environment [Zur93]. As this is a nonunitary process, one typically invokes Lindblad-type master equations to describe quantum systems subjected to decoherence. As we will just briefly talk about decoherence, I will not describe the full formalism. Instead, I will limit myself to describing the effect of particular decoherence processes on particular quantum system when that situation arises within the dissertation. One of the correlations that gets destroyed by decoherence is entanglement. Given the importance of entanglement for quantum information-processing tasks,

42 Chapter 2. Information Theory in the classical and the quantum world 24 the preservation of entanglement is paramount to anyone trying to exploit quantum systems to perform tasks not feasible classically. Unfortunately entanglement displays a very unsettling behaviour when subjected to decoherence. While one might expect correlations like entanglement to decay in an exponential way it turns out that entanglement does show, generally, a decay following a nonanalytical curve, which makes it vanish completely in a finite amount of time. This phenomenon, dubbed entanglement sudden death (ESD) [YE09, Col10], might pose specific challenges for the development of robust quantum information protocols, but it is not really surprising in the view of the geometry of separable states, since separable states have nonzero measure in the space of all states [ŻHSL98]. In a decoherence process that involves decay to a separable equilibrium state that does not lie on the boundary between separability and entanglement, the decohering state will cross that boundary before reaching the equilibrium state. 2.4 Maxwell s demon and Landauer s principle Maxwell proposed the following Gedankenexperiment: Consider an isolated box filled with some gas in thermal equilibrium. The box is divided into two sections connected by a trapdoor. An intelligent being, a demon, sits in the box and can open and close the trapdoor. In particular he can see the single gas molecules. Maxwell supposed that such a demon could in principle lower the system s entropy by looking at the molecules that fly towards the trapdoor and opening or closing the trapdoor so that slower molecules collect on one side of the box and faster molecules on the other side. Maxwell intended to show up the limitations of the Second Law of Thermodynamics and to emphasize its probabilistic nature. Maxwell s idea was debated for over 100 years in an effort to save the Second Law, thus leading to interesting connections between thermodynamics and information theory [LR02]. The resolution generally accepted today was put forward by Bennett [Ben82] and is based on Landauer s principle. Landauer showed that the erasure of information is tied to an increase in entropy [Lan61]. This is usually

43 Chapter 2. Information Theory in the classical and the quantum world 25 referred to as Landauer s principle today. The minimal cost for the erasure of one bit of information is kt ln 2. Bennett argued that Maxwell s demon acquires information when measuring the particle s velocities in order to operate the trapdoor. As the system we are considering is isolated and the demon only has a finite memory, he needs to erase his memory periodically. The entropy increase from erasure balances the reduction of entropy from the separation of fast and slow particles. In a famous paper of 1929 [Szi29], Szilárd envisioned a different incarnation of Maxwell s demon: Consider a cylinder containing only one particle in contact with a heat bath (the walls of the cylinder). A divider could be inserted, separating the cylinder into two equal parts. Upon acquiring the information in which part the particle is, a demon could insert a piston into the empty part, remove the divider, and let the particle isothermally push the piston outwards, doing some work W on a load coupled to the piston. The heat bath transfers energy Q=W to the particle in the process. After the particle pushes the piston out, it has the same volume accessible to it as in the beginning of the process. The heat bath having transferred energy to the particle, which does work, this cyclic process becomes a conversion of heat to work, in violation of the Second Law. The resolution to this paradox again involves accounting for the information acquired by the demon, which has to be erased at some point, given a finite memory, in order for the demon to be ready for a new cycle of the process. Interestingly Szilárd introduced the modern concept of information and used what was later called a bit by Shannon, as the demon has to acquire one bit of information in order to extract work. The demons we will be taking about in Chap. 4 will be less ambitious than Maxwell s and Szilárd s. They won t try to challenge the Second Law of Thermodynamics, but work in accordance with it to transform a low entropy system to a high entropy system while extracting work and then will move on to another copy of the system. We will be interested in the net work these demons can extract on average keeping close account of the cost they have to pay for the erasure of information in the process. This concludes the brief introduction to (quantum) information theory. The

44 Chapter 2. Information Theory in the classical and the quantum world 26 focus of this chapter was to provide the background necessary to follow Chaps. 4 and 5 of this dissertation for a more detailed discussion of the ideas presented above see [NC00].

45 27 Chapter 3 Discord and other nonclassical correlations beyond entanglement 3.1 Introduction At the heart of quantum information theory lies the objective to pinpoint the features that make it distinct from a classical theory. A complete understanding of these features would help us exploit this nonclassicality to perform information-processingtasks that are not possible with a classical system. One obvious feature that makes a quantum system distinct from a classical system is the kind of correlations, having no classical counterpart, that can be shared by several quantum systems. Entanglement is one type of nonclassical correlation which has proven to be a crucial resource in several quantum information-processing-tasks. Separable states, however, can have nonclassical correlations even though they are unentangled. Bennett et al. [BDF + 99] found an orthogonal set of product states, that cannot be reliably distinguished by separate observers. As a mixture of these states is not entangled, this discovery hinted at the fact that some other type of correlation, different from entanglement, has to account for a distinctly nonclassical behavior of some systems.

46 Chapter 3. Discord and other nonclassical correlations beyond entanglement 28 A variety of measures have been proposed in an attempt to quantify nonclassical correlations for bipartite systems [OZ01, OHHH02, Luo08a, PCMH09, WPM09, BT10, LCS11, MPS + 10] that can be nonzero for separable, but nonclassical states. In this chapter I will mainly focus on Quantum Discord, which is the quantity that attracted the most attention. In fact, sometimes all quantum correlations other than entanglement get lumped together under the name of quantum discord. I will try to be precise and use the name Quantum Discord, or simply just discord, only when referring to the quantity originally proposed by Ollivier and Zurek [Zur00, OZ01] and independently by Henderson and Vedral [HV01]. This section will not be comprehensive; I will only point out some results that are either of general interest or needed in the later chapters. A structured approach to the definition of these types of correlation measures is the focus of the next chapter, while a more complete account on the properties and various other results related to these correlation measures can be found in Modi et al. s review article [MBC + 12]. 3.2 Discord Ollivier and Zurek conceived quantum discord as a measure of disagreement between two forms of mutual information, S(X : Y ) and J(Y X), that are equivalent classically. S(X : Y ) is the mutual information from Eqs. (2.9) and (2.28), while J(Y X) is based on the identity for classical entropies: H(X : Y ) = H(X) H(X Y ). (3.1) They noted that in order to generalize the expression H(X Y ) to the quantum case one needs to specify a measurement basis in which information about system Y was obtained. Given a set of one-dimensional projectors {Π y } on system Y, the state of the combined system ρ X y after obtaining measurement outcome y is ρ X y = Π yρ X,Y Π y p y, (3.2) with p y = tr(π y ρ X,Y ).

47 Chapter 3. Discord and other nonclassical correlations beyond entanglement 29 Now Ollivier and Zurek [Zur00, OZ01] argued that an appropriate form for quantum version of the conditional entropy is p y S(ρ X y ) H {Πy }(X Y ), (3.3) y and used this to define S {Πy }(X : Y ) in analogy to Eq. (3.1): S {Πy }(X : Y ) = S(X) H {Πy }(X Y ). (3.4) This quantity is identical to the quantity that Henderson and Vedral [HV01] identified with the amount of classical correlations that can be extracted from a quantum system through the measurement {Π y }. As a measure that characterizes all the classical correlations accessible in a quantum state, Henderson and Vedral proposed to maximize this quantity over all measurements. Intuitively a measure of quantum correlations is the difference between the total correlations as measured by S(X : Y ) and the classically accessible correlations measured by max {Πy } S {Πy }(X : Y ) J(Y X). This is the quantity that Ollivier and Zurek dubbed quantum discord D(Y X): D(Y X) = S(X : Y ) J(Y X). (3.5) The arrow emphasises that this quantity is inherently asymmetric, as only one of the systems here is being measured. As is desirable for a correlation measure that is supposed to measure the truly quantum correlations, this quantity turns out to be non-negative 1. Since it is based on entropic quantities, it is invariant under local unitary transformations, which intuitively should not change correlations between two systems. For the case of pure states, it is identical to the entanglement measures The intriguing part about discord however is that in the realm of mixed states, it tries to capture more correlations than just entanglement. An example of a two qubit 1 The Appendix contains a proof of the non-negativity of discord, but I urge the reader to wait until the next chapter before looking at the proof. We will slightly adapt the notation and introduce a second measurement when defining discord. While the resulting quantity is the same, it is seen from a different perspective.

48 Chapter 3. Discord and other nonclassical correlations beyond entanglement 30 state ρ disc that is not entangled, but has nonzero discord can be found in [DacVB10]: ρ disc = 1 ( ). (3.6) Over the last 10 years discord has received a great deal of attention. Unfortunately, a big part of the research devoted to discord is not well motivated, because this quantity has no intrinsic operational meaning attached to it. While some progress has been made in providing operational interpretations [MBC + 12], many of these attempts are arguably contrived. In the following, I will briefly describe the operational interpretation for discord, that I, personally, find the most convincing. Here discord is linked a quantum communication protocol [MD11, MD13]. Suppose Alice and Bob share a bipartite quantum state. Alice s goal is to transfer her part of the state to Bob. To accomplish this task she can use local operations and classical communication (LOCC) as a classical resource as well as shared entanglement as a quantum resource. We will assume LOCC comes for free and focus on the quantum resource needed to achieve their goal. One way to merge these states is for Alice to use the quantum teleportation protocol [BBC + 93]. It turns out, however, that she can do better and only spend S(A B) bits of entanglement to accomplish her goal [HOW05]. Moreover, this protocol, called quantum state merging, also provides a suggestive interpretation of what it means that S(A B) can be negative. In that case, Alice does not have to spend entanglement bits to merge the states, but rather recovers ebits in the process. If Bob measures his part of the system before he receives Alice s part, the cost for the state merging process will increase: Bob will destroy some correlations during his measurements, which Alice otherwise could have take advantage of. Madhok and Datta [MD11] showed that the amount the state merging cost increases when Bob measures his system first is quantified by the discord D(B A) of the original state. This measurement could also be thought of as a decoherence process, where the environment measures Bob s system and destroys correlations [MD13].

49 Chapter 3. Discord and other nonclassical correlations beyond entanglement Other forms of nonclassical correlations In 2003 Zurek [Zur03] showed that the advantage a nonlocal quantum demon has over his local classical counterparts in extracting work from a quantum system can be quantified through a form of discord. Unfortunately with this he started a trend in attaching the same name, discord, to different quantities that share some common traits. To emphasise that the quantity involved in the discussion of demons is different form the original discord, we will refer to it as demon discord. Another quantity that is similar to, but different from discord was proposed by Luo in 2008 [Luo08a]. The idea behind this quantity, which Luo called measurementinduced disturbance (MID), is to quantify how much correlation in a system gets disturbed when the system is measured. Unlike discord this quantity is symmetric under exchange of the two systems. An advantage over discord is that it is easier to compute. The measurements involved here are fixed to be performed in the eigenbasis of the marginal density operators, so unless the marginal states are degenerate, no optimisation procedure needs to be performed. These are just two examples of a plethora of similar quantities [MPS + 10] that try to capture more quantum correlations than entanglement does. We will meet some more of these measures in the next chapter and show how they are related. For now we will focus on a more general point, i.e., the set of states for which all these correlation measures vanish. 3.3 The set of classical states A state is considered to be classical when its density matrix ρ cc is diagonalized in a basis that is the product of basis elements for each subsystem [Luo08a]: ρ cc = a,b p a,b e a e a f b f b. (3.7) A state written in this form is characterized by purely classical correlations. Moreover we can measure the state without disturbing it if we perform the measurements in

50 Chapter 3. Discord and other nonclassical correlations beyond entanglement 32 basis { e a } for the first system and basis { f b } for the second system. One kind of correlation measure, necessarily symmetric, will be, zero if and only if the joint state can be written in form of Eq. (3.7). The set of states for which discord vanishes is a bit bigger than that. In fact, if we consider system A being measured, in order for the joint state to have zero discord, one needs to be able to diagonalize the state ρ cq in a conditional product basis [Dat10]: ρ cq = a,b p a,b e a e a f b a f b a, (3.8) This means that the state is block-diagonal in a product basis: ρ cq = a p a e a e a ρ B a. (3.9) In this case only system A is undisturbed by an appropriate local measurement. Hence, these latter states ρ cq are sometimes called classical-quantum, whilst the former ρ cc are dubbed classical-classical. Clearly the classical-classical states are a subset of the classical-quantum state, which in turn are a subset of the separable states ρ sep for Eq. (2.31). Ferraro et al. [FAC + 10] pointed out a big conceptual difference between the set of separable states and the set of classical-quantum states: They proved that the set of states with zero discord has measure zero in the space of all states and is nowhere dense. As a consequence a randomly selected bipartite quantum state will have non-zero discord. Moreover, a decohering state will not cross this classicality boundary before reaching an equilibrium state. As a consequence, there will be no phenomenon like entanglement sudden death for discord. In Chap. 5, we will see a visualisation of this behaviour. While the state space we consider there will be restricted, it is well suited to demonstrate the points above. Moreover, we will see that unlike the set of separable states, the set of classical states is not convex. While this chapter reported some properties on nonclassical correlations and presented discord in its original form, we will take on a slightly different perspective in the next chapter. There we take a more logical approach to nonclassical correlations and define a framework from which several measures on noncalssical correlations, one of them being quantum discord, emerge.

51 33 Chapter 4 A framework for entropic measures of nonclassical correlations In this chapter we unify some of the measures discussed in Chap. 3 in a single framework, with the goal of clarifying the relations between these several quantities. Some ordering relations will follow naturally from the formulation of our framework; others we will have to prove explicitly. 4.1 Introduction Maxwell demons observe a physical system and use the information obtained to extract work from the system [Max91]. For multipartite systems, we can distinguish quantum Maxwell demons, which have knowledge of the entire density operator and can manipulate and make measurements on the joint system, from classical demons, which can only perform operations and make measurements on the subsystems of the multipartite system. Because a single classical demon cannot be everywhere at the same time, it must recruit local demons to gather, process, and use information about the local systems; thus it is better to think of a classical demon as a collection of local demons. These local demons might or might not be allowed to communicate with each other using classical channels. The amount of work that the two kinds

52 Chapter 4. A framework for entropic measures of nonclassical correlations 34 of demons, quantum and classical, can extract from a given multipartite quantum state by employing protocols within each demon s means is a way of comparing quantum-information-processing protocols with classical ones. This demonology [Zur00, OHHH02, Zur03, BT10] is but one of several attempts [Zur00, OZ01, HV01, RR02, Zur03, PHH08, Luo08a, WPM09, BT10, MPS + 10] to track down and quantify the correlations that exist in multipartite quantum states. The nonclassical part of these correlations is not just quantum entanglement, even though entanglement is a part of it. The open question of pinning down why mixedstate quantum algorithms can solve certain problems exponentially faster than the best known classical ones [JL03], even in the absence of any significant entanglement, is one of the main motivations behind studying the nonclassical correlations in quantum states other than entanglement [DV07, DSC08, Dat08, DG09, Eas10]. We consider only bipartite states in this work. For our numerical work, the discussion is specialized yet further to states of two qubits. Correlations between systems can be quantified in terms of correlation coefficients and covariance matrices or in terms of entropic measures like mutual information. We choose the latter approach as the preferred one in information theory. The aim of this work is to formulate a framework in terms of which the several entropy-based measures of nonclassical correlations that have been proposed can be classified and understood. Constructing the framework leads to two new measures we have not seen previously in the literature. The focus here is not so much on unifying various measures, as in Ref. [MPS + 10], but rather on clarifying the relationships among them. The setting for our framework is two systems, A and B, with a joint quantum state ρ AB. We consider three types of nonclassical-correlation measures, M(ρ AB ), between A and B: 1. Mutual-information-based measures. 2. Conditional-entropy-based measures. 3. Demon-based (joint-entropy-based) measures.

53 Chapter 4. A framework for entropic measures of nonclassical correlations 35 The type-2 correlation measures can be asymmetric between A and B because conditional entropy is typically asymmetric. As Landauer pointed out, when talking about demons, erasure of the demon s memory and the associated thermodynamic cost is an essential feature for assessing what a demon can do [Lan61]. As we mentioned above, a classical demon that works on a bipartite quantum system is best thought of as two local demons working in concert. Whether the two demons can communicate impacts their ability to coöperate. So the demon-based measures are thus further divided into two classes: i. Erasure without communication between the demons. ii. Erasure with communication between the demons. All the measures of nonclassical correlations we consider here are constructed as the difference between a quantum entropic measure, Q(ρ AB ), and its classical counterpart, C(ρ AB ), which is derived from the probabilities for results of local measurements on one or both of the subsystems. The thinking behind this construction is that Q quantifies some notion of all the correlations in the system, whereas the corresponding classical C captures only the corresponding classical correlations. The difference, M = Q C, is therefore a way of quantifying the nonclassical correlations in the quantum state. The results of local measurements are all that local classical observers (demons) can access, and these measurement results are used to probe the correlations (if any) between A and B. We do not want, however, our measure of nonclassical correlations to depend on the specifics of the measurement performed. Hence, in its construction, the classical measure, C(ρ AB ), is maximized over all possible measurements within specific measurement strategies that are defined beforehand. In some instances, when maximization is necessary, we are able to show that the maximum is attained on rank-one POVMs; in other cases, we restrict the maximization to rank-one POVMs. We give a full discussion of these different situations and the issues surrounding rank-one POVMs after we have developed our framework.

54 Chapter 4. A framework for entropic measures of nonclassical correlations 36 We thus imagine that there are classical observers A and B demons or otherwise who have access to the two parts of the bipartite system. We allow these observers to employ one of three measurement strategies: a. Local, rank-one-projector measurements in the eigenbases of the marginal density operators. b. Unconditioned local measurements. c. Conditioned local measurements. For strategy (a), the local measurements are unique modulo degeneracies in the marginal density operators. The other two strategies require maximization of the classical measure C over the measurements allowed by the strategy. The first two measurement strategies do not require the observers to communicate with each other, but the last one does. Consequently, the first two strategies are symmetric between A and B. For the third strategy, A performs a measurement and communicates the result to B, who can then condition his measurement on the result communicated by A. This makes the nonclassical correlation measures that are based on the third measurement strategy asymmetric between A and B. We now have three types of correlation measures and three measurement strategies, and we can label the resulting correlation measures with the type and the strategy. For example, M 1b refers to the nonclassical correlation measure constructed as the difference between quantum and classical mutual informations, where unconditioned local measurements are used to construct the classical mutual information. There is a natural hierarchy in the three types of measurements strategies. Allowing arbitrary, unconditioned local measurements, as in strategy (b), is a restriction of the conditioned local measurements of strategy (c), since to get (b) from (c), observer B simply chooses to ignore any communication A might have sent regarding her measurement results. Likewise, measuring in the local eigenbases of the marginal density operators, as in strategy (a), is a restriction of the arbitrary, unconditioned local measurements of strategy (b). Thus, when we maximize over the measurements

55 Chapter 4. A framework for entropic measures of nonclassical correlations 37 in a particular strategy, the classical measure C cannot decrease and generally it increases as we move from (a) to (b) to (c). This is saying that the more general the measurements the local observers are allowed to do, the more they can expect to discover about any classical correlations that exist between the subsystems. Since our nonclassical-correlation measure M is the difference between Q and C, M cannot increase and generally it decreases as we move from (a) to (b) to (c), i.e., M ja M jb M jc for j = 1, 2, 3. In Sec. 4.2 we formulate our framework: Sec reviews the bipartite entropic information measures that we use in constructing our framework; Sec spells out the description of local measurements for strategies (a) (c); Sec defines the nonclassical-correlation measures and discusses relations among them; and Sec considers the issues raised by assuming the local measurements are described by rankone POVMs and also whether one can specialize further to measurements described by rank-one projectors. In Sec. 4.3 we present numerical results comparing the various measures for two-qubit states, assuming that the local measurements can be described by orthogonal rank-one projection operators. App. A provides additional information. 4.2 Framework for entropic measures of nonclassical correlations In this section we develop our framework for measures of nonclassical correlations and explore properties of the various measures the framework leads to Entropic measures of information and correlation Entropic measures of information quantify how much information can be extracted from a system or, more poetically, how much information is missing about the fine-grained state of the system.

56 Chapter 4. A framework for entropic measures of nonclassical correlations 38 Figure 2.1 is a useful pictorial representation of the relationships among the entropies and entropic measures of correlation that apply to bipartite systems. The figure provides an accurate representation for classical entropies. In the quantum case, some of the quantities cannot be represented or are misrepresented by this diagram, but even so, the diagram is a useful tool because it captures correctly the relationships among the various entropies. For a bipartite state ρ AB of systems A and B, the quantum entropic quantities that will be used in the ensuing discussion are the following: 1. S(A, B) = S(ρ AB ) = tr(ρ AB log ρ AB ), the joint von Neumann entropy of the whole system. 2. S(A) = S(ρ A ) = tr A (ρ A log ρ A ) and S(B) = S(ρ B ) = tr B (ρ B log ρ B ), the von Neumann entropies of the marginal density operators. 3. S(B A) = S(A, B) S(A) and S(A B) = S(A, B) S(B), the quantum conditional entropies. 4. S(A : B) = S(A) + S(B) S(A, B), the quantum mutual information, which is related to the quantum conditional entropies by S(A : B) = S(B) S(B A) = S(A) S(A B). The quantum mutual information can also be written as a quantum relative entropy, S(A : B) = S(ρ AB ρ A ρ B ), (4.1) where the relative entropy is defined by S(ρ σ) = S(ρ) tr(ρ log σ). (4.2) Local measurements on the bipartite quantum system are described by a joint probability distribution p ab for outcomes labeled by a and b. Bayes s theorem relates the joint, conditional, and marginal distributions: p b a p a = p ab = p a b p b. These distributions are used to define the classical information measures:

57 Chapter 4. A framework for entropic measures of nonclassical correlations H(A, B) = H(p ab ) = a,b p ab log p ab, the Shannon entropy of the joint distribution p ab. 2. H(A) = H(p a ) = a p a log p a and H(B) = H(p b ) = a p b log p b, the Shannon entropies of the marginal distributions, p a and p b. 3. H(B A)=H(A, B) H(A) = a p ah(b a) and H(A B)=H(A, B) H(B) = a p bh(a b), the classical conditional entropies. H(B a) = b p b a log p b a and H(A b) = a p a b log p a b are the Shannon entropies of the conditional distributions p b a and p a b ; the conditional entropies are averages of H(B a) over p a and H(A b) over p b. 4. H(A : B) = H(A) + H(B) H(A, B) = a,b p ab log(p ab /p a p b ), the classical mutual information. H(A : B) is the relative information of the joint distribution p ab with respect to the product of the marginals, p a p b, H(A : B) = H(p ab p a p b ) ; (4.3) the classical relative information, which is always nonnegative, is defined by H(p j q j ) = j p j log(p j /q j ) = H(p j ) j p j log q j. (4.4) We also have H(A : B) = H(B) H(B A) = H(A) H(A B). Figure 2.1 in Chap. 2 summarizes the relations among the classical entropies; it works because the classical conditional entropies and the classical mutual information are all nonnegative. This leads to several inequalities that can be read off Fig For example, we can see that max ( H(A), H(B) ) H(A, B) H(A) + H(B). (4.5) The lower bound on H(A, B) is saturated when knowing one subsystem completely determines the other (the two circles in Fig. 2.1 are either identical or become nested), i.e., H(A : B) = min ( H(A), H(B) ). The upper bound is saturated when there are no correlations between A and B, i.e., H(A : B) = 0, so determining one subsystem gives

58 Chapter 4. A framework for entropic measures of nonclassical correlations 40 no information about the other (the two circles in Fig. 2.1 are disjoint). For quantum entropies the lower bound in Eq. (4.5) does not hold, which is equivalent to saying the quantum conditional entropies can be negative. The simplest counter-example is a two-qubit Bell state: the joint state is pure and, hence, has zero entropy, but the marginal states are completely mixed, so their entropies are maximal and both equal to one Local measurements We now spell out the general description of the local measurements that applies to measurement strategies (a) (c). Although we only need measurement statistics and, hence, only need POVMs to evaluate the classical entropic measures, we start our description with quantum operations, partly to be general and partly so we can deal with post-measurement states in a subsequent discussion of Maxwell demons. The measurement on A is described by quantum operations [NC00] that are labeled by the possible outcomes a of the measurement on A: A a = α A aα A aα. (4.6) The quantum operation is applied to a density operator by inserting the density operator in place of the. The operators A aα, the Kraus operators of A a, combine to give the POVM element for outcome a, E a = α A aαa aα, (4.7) and the POVM elements satisfy a completeness relation, I A = a E a. The absence of communication in strategies (a) and (b) makes them quite straightforward. The measurement on B is described by a set of quantum operations, B b = β B bβ B bβ. (4.8) These give POVM elements F b = β B bβ B bβ, (4.9)

59 Chapter 4. A framework for entropic measures of nonclassical correlations 41 which satisfy a completeness relation I B = b F b. The state of the joint system after measurements with outcomes a and b is ρ AB ab = A a B b (ρ AB )/p ab, where p ab = tr ( A a B b (ρ AB ) ) = tr(e a F b ρ AB ) (4.10) is the joint probability for outcomes a and b. The post-measurement joint state and the joint probability marginalize to the subsystems in the standard way. We need to be more careful with strategy (c) because of the communication from A to B. We handle strategy (c) in a general way that allows us to interpolate between (b) and the extreme case of (c) in which every outcome a leads to a different measurement on B. We do this by introducing a set C whose elements c label the possible measurements to be made on B. We let A stand for the set of outcomes a, and we define a function c(a) that maps an outcome a to the corresponding value in C. We let A c = {a c(a) = c} be the subset of A that leads to the B measurement labeled by c. The subsets A c partition A into disjoint subsets. We can regard C as another variable in our analysis; it is a coarse graining of the measurement on A. Formally, we have that C is perfectly correlated with A, i.e., p c a = δ c,c(a), implying that H(C A) = 0 and H(A : C) = H(C). Should there be only one possible measurement on B, i.e., only one value of c, then there is no communication, and the situation reduces to strategy (b). The extreme case of (c) corresponds to having a different value of c for each outcome a, in which case there is no difference between the outcome set A and the set C. The state of the joint system after the measurement on A yields outcome a is ρ AB a = A a (ρ AB )/p a, where p a = tr ( A a (ρ AB ) ) = tr A (E a ρ A ) (4.11) is the probability for outcome a. The state of system B, conditioned on outcome a, is ρ B a = tr A (ρ AB a ) = tr ( ) A Ea ρ AB ; (4.12) p a notice that this is determined by the POVM element E a. The probability for making

60 Chapter 4. A framework for entropic measures of nonclassical correlations 42 measurement c on B follows formally from p c = a p c a p a = a A c p a = tr A (E c ρ A ). (4.13) Here we introduce coarse-grained POVM elements for the measurement on A, labeled by the measurement to be made on B: E c = a A c E a. (4.14) Notice that if there is only one possible measurement on B, i.e., only one value of c, then E c = I A ; when there is a different measurement for each outcome a, the POVM elements E c are the same as the POVM elements E a. We also have the state of B conditioned on the coarse-grained outcome c: ρ B c = tr ( ) A Ec ρ AB. (4.15) p c Notice that Eqs. (4.12) and (4.15) imply that ρ B = a p a ρ B a = c p c ρ B c. (4.16) We turn our attention now to the measurements on B. We let B stand for the set of all outcomes on B for all the possible measurements on B. We define a function c(b) that maps an outcome b to the measurement c in which it occurs, and we define B c = {b c(b) = c} to be the subset of B outcomes for the measurement labeled by c. The subsets B c partition the set of all possible outcomes on B into disjoint subsets. We again have perfect correlation, i.e., p c b = δ c,c(b), implying that H(C B) = 0 and H(B : C) = H(C). The measurement on B that is labeled by c is described by quantum operations B b c = β B bβ c B bβ c, (4.17) The Kraus operators give the POVM elements for this measurement, F b c = β B bβ c B bβ c, (4.18)

61 Chapter 4. A framework for entropic measures of nonclassical correlations 43 and these satisfy a completeness relation I B = b B c F b c. In sums over b, we can let the sum run over the outcomes of all the possible measurements on B by the artifice of defining B bβ c = 0 for b / B c and, hence, F b c = 0 for b / B c. where The state of the joint system, conditioned on outcomes a and b, is ρ AB ab = A a B b c(a) (ρ AB ) p ab = B b c(a)(ρ AB a ) p b a, (4.19) p ab = tr ( A a B b c(a) (ρ AB ) ) = tr(e a F b c(a) ρ AB ) = p a tr B (F b c(a) ρ B a ) (4.20) is the joint probability for a and b and p b a = tr ( B b c(a) (ρ AB a ) ) = tr(f b c(a) ρ B a ) (4.21) is the conditional probability for b given a. Notice that p ab and p b a are nonzero only if b B c(a) or, equivalently, only if a A c(b). For our purposes, it is easier to work with the coarse-grained outcomes c, which specify the measurements on B. Indeed, the joint probability for b and c is p bc = a p c ab p ab = a A c p ab = tr(e c F b c ρ AB ) = p c tr B (F b c ρ B c ). (4.22) Notice that p bc is nonzero only if b B c. Thus the conditional probability of b given c takes the form p b c = p bc p c = tr B (F b c ρ B c ), (4.23) and the unconditioned probability for b is p b = c p bc = tr(e c(b) F b c(b) ρ AB ) = p c(b) tr B (F b c(b) ρ B c(b) ). (4.24) Measures of nonclassical correlations In this subsection we formulate our framework for entropic measures of nonclassical correlations, considering in turn the three types of measures introduced in Sec. 4.1

62 Chapter 4. A framework for entropic measures of nonclassical correlations 44 and for each type, the three local measurement strategies, (a), (b), and (c). For strategy (a), the local measurements are in the eigenbases of the marginal density operators. For strategies (b) and (c), we assume that the measurements are described by rank-one POVMs, which means that E a and F b c are multiples of rank-one projection operators. We discuss this assumption in Sec To compare and relate the various measures, we rely on two inequalities that relate the quantum and the classical entropies: the POVM inequality (see App. A.1 for a proof) and the ensemble inequality [NC00]. The POVM inequality relates the quantum entropy for a state ρ to the classical entropy for probabilities p j = tr(e j ρ) obtained from (nonzero) POVM elements E j : H(p j ) + j p j log(tre j ) = j ( ) pj p j log S(ρ). (4.25) tre j A rank-one POVM is one such that all the POVM elements are rank-one, i.e., E j = µ j P j, where P j is a rank-one projection operator and 0 µ j = tre j 1. The trace of the completeness relation implies that j µ j = (dimension of the quantum system). For a rank-one POVM, we have H(p j ) S(ρ) j p j log µ j S(ρ). (4.26) The ensemble inequality [NC00] says that the Shannon information of a set of ensemble probabilities q j exceeds the Holevo quantity of the ensemble: ( ) H(q j ) S q j ρ j j j q j S(ρ j ). (4.27) For strategy (a), where the local measurements are in the eigenbases of the marginal density operators, we have immediately that H(A) = S(A) and H(B) = S(B). For both (b) and (c), we can apply the POVM inequality in its rank-one form to p a = tr(e a ρ A ) to conclude that H(A) S(A). Similarly, for strategy (b), the POVM inequality applied to p b = tr(f b ρ B ) gives H(B) S(B). For strategy (c), we need a

63 Chapter 4. A framework for entropic measures of nonclassical correlations 45 chain of inequalities to conclude that H(B) S(B): H(B) = H(C, B) = H(C) + H(B C) = H(p c ) + c p c H(B c) H(p c ) + p c S(ρ B c ) (4.28) c ( ) S p c ρ B c (4.29) c = S(ρ B ) = S(B). (4.30) The first inequality (4.28) is a consequence of applying the POVM inequality to Eq. (4.23), the second inequality (4.29) is an example of the ensemble inequality, and the final equality uses Eq. (4.16). Type 1: Mutual-information-based measures For type-1 measures, we choose Q 1 = S(A : B) and C 1 = H(A : B), giving the difference measure M 1 = S(A : B) H(A : B). (4.31) We now apply the three measurement strategies introduced in Sec. 4.1 to obtain the classical mutual information H(A : B); this leads to three different type-1 measures. For strategy (a), the local measurements are made in the eigenbases of the marginal density operators, and this gives a nonclassical-correlation measure that we denote by M 1a. If the marginal density operators have nondegenerate eigenvalues, the marginal eigenbases are unique; in the case of degeneracy, one needs to maximize H(A : B) over the rank-one, projection-valued measurements in the degenerate subspaces to get a unique measure M 1a. The measure M 1a was introduced by Luo in [Luo08a] and called there the measurement-induced disturbance (MID). The same measure, in a different guise, had been proposed by Rajagopal and Randall in [RR02]; they defined what they called the quantum deficit as H(A, B) S(A, B), where H(A, B) is obtained from measurements in the marginal eigenbases. The quantum deficit and

64 Chapter 4. A framework for entropic measures of nonclassical correlations 46 MID are the same because they differ by the terms H(A) S(A) and H(B) S(B), which are zero for measurements in the marginal eigenbases. When strategy (b) is used, we obtain the measure M 1b = S(A : B) max H(A : B), (4.32) (b) where the classical mutual information has to be maximized over the unconditioned local measurements of strategy (b). The maximum classical mutual information was introduced in [PHH08] as a measure of classical correlations, and the same paper suggested M 1b as a measure of nonclassical correlations. This measure was investigated in detail by Wu, Poulsen, and Mølmer (WPM) in [WPM09], and we refer to it as the WPM measure, while denoting it as M 1b. The optimal unconditioned local measurements are not necessarily orthogonal-projection-valued. An example of a case in which the maximization requires POVMs and not just projective measurements was given in [WPM09]; we review and extend this example in App. A.3. In addition, the optimal local measurements do not generally occur in the marginal eigenbases, which implies that M (MID) 1a M (WPM) 1b. For strategy (c), the classical mutual information H(A : B) can be made arbitrarily large, thus allowing M 1c to be arbitrarily negative. This is easy to see by considering the extreme case of (c) in which every outcome a leads to a different measurement on system B; then, as noted in Sec , H(A : B) = H(A), which can be as big as desired by giving the measurement on A an arbitrarily large number of outcomes. We conclude that M 1c has nothing to do with quantifying nonclassical correlations, so we drop M 1c from our array of possible measures. Type 2: Conditional-entropy-based measures For type-2 measures, we choose Q 2 = S(B A) and C 2 = H(B A). The result is the difference measure M 2 = H(B A) S(B A). (4.33)

65 Chapter 4. A framework for entropic measures of nonclassical correlations 47 We notice immediately that M 2 = M 1 + [ H(B) S(B) ] M 1. (4.34) This shows that a type-1 measure is always less than or equal to the type-2 measure that uses the same measurement strategy, with equality only when B is measured in the marginal eigenbasis. Measurements in the eigenbases of the marginal density operators have H(B) = S(B), so for strategy (a), we have M 2a = M 1a, and our measure is again MID. Strategy (b) gives the measure M 2b = min H(B A) S(B A), (4.35) (b) where we have to minimize H(B A) over all unconditioned local measurements. We can conclude from general considerations that M (MID) 1a = M (MID) 2a M 2b M (WPM) 1b. Notice also that the unconditioned local measurements that minimize H(B A) need not be the same as those that minimize H(A B). This means that M 2b is intrinsically asymmetric between subsystems A and B even though the measurement strategy is symmetric. Strategy (c) gives the measure M 2c = min H(B A) S(B A). (4.36) (c) The POVM inequality immediately gives a bound on H(B A), H(B A) = a p a H(B a) a p a S(B a) H {Ea }(B A). (4.37) When we are allowed to make conditional measurements on B, the bound can be achieved by measuring B, for outcome a, in the eigenbasis of ρ B a. Hence, with the conditional measurements on B specified, the minimization of the classical conditional entropy, H(B A), is reduced to choosing a measurement on A that minimizes the conditional entropy H {Ea }(B A): min (c) H(B A) = min H {E {E a } a }(B A) H(B A). (4.38)

66 Chapter 4. A framework for entropic measures of nonclassical correlations 48 The quantity H(B A) is a special sort of classical conditional entropy. The resulting measure is equivalent to the quantum discord [Zur00, OZ01] introduced in Chap. 3: M 2c = H(B A) S(B A) D(A B). (4.39) In App. A.3, we exhibit joint states that show that to find the minimum H(B A) and, hence, to find the quantum discord sometimes requires rank-one POVMs, not just orthogonal-projection-valued measurements. Notice that in Chap. 3 we introduced discord as a difference between mutual information-like quantities in accordance with the original definition by Ollivier and Zurek [OZ01]: D(A B) = S(A : B) J(A B). This is equivalent to Eq. (4.39), as one can readily subtract S(B) from S(A : B) and J(A B): (S(A : B) S(B)) (J(A B) S(B)) = S(B A) + H(B A). Moreover the definition of discord in Chap. 3 does not make use of conditioned measurements measurements on B, but rather assumed that the quantity to be minimized over measurements on A is the conditional entropy H {Ea }(B A). We can conclude from general considerations that M (MID) 1a = M (MID) 2a M 2b M (discord) 2c. Our present considerations do not, however, provide an ordering of the WPM measure and quantum discord. We return to the ordering of WPM and discord in Sec and show in App. A.2 that M (WPM) 1b M (discord) 2c. Type 3: Demon-based measures Type-3 measures quantify the difference in the work that can be extracted from a quantum system by quantum and classical demons. The demons extract work by transforming the initial joint state ρ AB to the fully mixed joint state using any means at their disposal, including measurements. We assume here that all states of the system have the same energy so that all the work that the demons extract arises from the entropy difference between the initial and final states of the system; it is natural to choose k B T ln 2 as the unit of work. Throughout this chapter, whenever we talk

67 Chapter 4. A framework for entropic measures of nonclassical correlations 49 about extractable work and erasure cost, we actually mean average work and average erasure cost. The maximum work that can be extracted by a quantum demon by any means is given by the entropy difference between the initial and final states, W q = log(d A d B ) S(A, B), (4.40) where d A and d B are the dimensions of the two subsystems. The demon could extract this amount of work by devising an optimal process that directly transforms the joint state ρ AB to the maximally mixed state. It could, instead, make a measurement in the joint eigenbasis of ρ AB, extract work log(d A d B ) as the post-measurement pure eigenstate is transformed to the maximally mixed state, and then pay a price S(A, B) to erase its memory of the S(A, B) bits acquired in the measurement. The demon would then be ready to pick up another copy of the system and repeat the process. In contrast to a quantum demon, a local, classical demon can only manipulate the subsystem in its possession. In Sec. 4.1 we introduced two cases for the local demons that are dealing with our bipartite system. In case (i) the two demons are not allowed to communicate with each other. In this case, the maximum amount of work demon A can extract from subsystem A is log d A S(A). This can be achieved by an optimal process that directly transforms the marginal state ρ A to the maximally mixed state or by measuring in the marginal eigenbasis, extracting work log d A as the post-measurement pure state is transformed to the maximally mixed state, and then erasing the S(A) bits of measurement record at cost S(A). Since demon B is in the same situation, the maximum work the two local demons can extract is W c = log(d A d B ) S(A) S(B). (4.41) The difference in the amount of work that can be extracted by the quantum and classical demons, called the work deficit [OHHH02, Zur03, BT10], is the quantum mutual information: W q W c = S(A) + S(B) S(A, B) = S(A : B) M 3(i). (4.42)

68 Chapter 4. A framework for entropic measures of nonclassical correlations 50 Brodutch and Terno [BT10] have noted that the work deficit in the case of erasure without communication between the local demons provides an operational interpretation of the quantum mutual information. In case (ii) the local demons can communicate their measurement results and thus reduce their cost of erasure. In particular, the demons make local measurements, which in accord with the assumptions of this section are described by rank-one POVMs and thus leave the two subsystems in pure states. They can then extract work W + = log d A + log d B (4.43) as their respective systems are transformed to the maximally mixed state. They must then erase their memories of the measurement record so they are ready to handle another copy of the joint state ρ AB. In the absence of communication, the total erasure cost is W = H(A) + H(B) S(A)+S(B), with the minimum attained for measurements in the marginal eigenbases; the net work the demons can extract is that of case (i), i.e., W c = W + W = log d A d b S(A) S(B). If the demons can communicate, however, as in case (ii), then they can take advantage of correlations between their measurement results to reduce their erasure cost to the joint classical information in their measurement records, W = H(A, B), which gives net work W c = W + W = log(d A d B ) H(A, B). (4.44) Thus in case (ii), the work deficit becomes W q W c = H(A, B) S(A, B) = M 3, (4.45) giving us joint-entropy-based measures of nonclassical correlations, with Q 3 = S(A, B), C 3 = H(A, B), and M 3 = Q C. We now have to consider the three measurement strategies for the local demons, but before embarking on that, we note that M 3 = M 2 + [ H(A) S(A) ] = M 1 + [ H(B) S(B) ] + [ H(A) S(A) ], (4.46)

69 Chapter 4. A framework for entropic measures of nonclassical correlations 51 so for each measurement strategy, we have M 3 M 2 M 1, as we have noted earlier. For strategy (a), measurement in the marginal eigenbases, we have H(A) = S(A) and H(B) = S(B), so we again get the MID measure, i.e., M 3a = M 2a = M 1a ; this is the form in which Rajagopal and Randall [RR02] defined what they called the quantum deficit. For strategy (b), we have to minimize H(A, B) over all unconditioned local measurements, M 3b = min H(A, B) S(A, B) ; (4.47) (b) in general, the result is not the same as M 2b or M 1b. For strategy (c), we have to minimize H(A, B) over all conditioned local measurements. The minimization over the conditioned measurements on B is simple, since as in Eq. (4.37), we have H(A, B) = H(A) + H(B A) H(A) + a p a S(B a) H {Ea }(A, B), (4.48) with equality if and only if the measurement on B, given outcome a, is in the marginal eigenbasis of ρ B a. Hence, with the conditional measurements on B specified, the minimization of the classical joint entropy, H(A, B), is reduced to choosing a measurement on A that minimizes the joint entropy H {Ea }(A, B): min (c) H(A, B) = min H {E {E a } a }(A, B) H(A, B). (4.49) The quantity H(A, B) is a special sort of classical joint entropy. The resulting measure of nonclassical correlations is M 3c = H(A, B) S(A, B). (4.50) This measure was hinted at in Zurek s original paper on discord [Zur00]. Ollivier and Zurek [OZ01] defined quantum discord as the quantity M 2c, but Zurek [Zur03] resurrected M 3c as a modified form of discord in his paper on discord and Maxwell demons. Brodutch and Terno [BT10] have also pointed out that M 3c is the measure that applies to demons that can communicate and use strategy (c) for their measurements. Hence, we can call M 3c the demon discord (dd).

70 Chapter 4. A framework for entropic measures of nonclassical correlations 52 As noted in Sec. 4.1, we have M MID 3a M 3b M (dd) 3c. Properties of nonclassical-correlation measures The following array neatly summarizes the measures of nonclassical correlations that we have found and the relations we have found among them: M (MID) 1a M (WPM) 1b = M (MID) 2a M 2b M (discord) 2c (4.51) = S(A, B) = M 3(i) M (MID) 3a M 3b M (dd) 3c The vertically oriented inequalities are best read by leaning your head to the left; in the absence of leaning, the wedges point toward the smaller quantity, as is standard. Of the potential measures we started with, the demon-based measure that assumes erasure without communication is special and gives the quantum mutual information. Of the remaining nine potential measures, we discarded one, M 1c, as meaningless; we found that the three measures in the left column of the array are all identical to the MID measure; we determined that three of the other measures are the WPM measure, quantum discord, and demon discord; and we are thus left with two new measures, M 2b and M 3b, although M 3b is very closely related to and perhaps identical to a discord-like measure introduced by Modi et al. [MPS + 10]. Modi et al. [MPS + 10] introduced a set of measures of quantum and classical correlations based on the relative-entropy distance (4.2) between a multi-partite state ρ and the nearest state σ ρ that is diagonal in a product basis, or between ρ and the nearest product state. The only one of these measures relevant to our discussion is their discord, which when specialized to bipartite states, is the distance D Modi = min σab S(ρ AB σ AB ), where σ AB is diagonal in a product basis. Modi et al. show that the minimum is attained on a state obtained by projecting ρ AB into a product basis, i.e., σ AB = a,b e a, f b e a, f b ρ AB e a, f b e a, f b, in which case,

71 Chapter 4. A framework for entropic measures of nonclassical correlations 53 S(ρ AB σ AB ) = S(σ AB ) S(ρ AB ). Thus we have D Modi = min { e a,f b } S(σ AB) S(ρ AB ). (4.52) Since S(σ AB ) is the classical joint entropy of a measurement made on ρ AB in the product basis e a, f b, this would be the same as our M 3b if we knew that the optimal local measurements for M 3b were described by orthogonal rank-one projectors. Brodutch and Terno [BT10] define three kinds of discord : their D 1 is the standard discord M (discord) 2c ; their D 2 is the demon discord M (dd) 3c ; and their D 3 is a discord-like quantity that uses a different conditional measurement strategy. This strategy allows conditioned local measurements, but with the measurement on A constrained to be in the marginal eigenbasis of ρ A. The Brodutch-Terno measurement strategy is a restriction of strategy (c), and (a) is a restriction of the Brodutch-Terno strategy. Measures based on it could thus be placed in the array (4.51) as an alternative intermediate column whose ordering with strategy (b) is indeterminate. All the measures in the array (4.51), except the quantum mutual information, are bounded above by MID, and MID is bounded above by the quantum mutual information. Similarly, MID, M 2b, and M 3b are bounded below by both the WPM measure and the quantum discord, and the demon discord M 3c is bounded below by discord. The WPM measure and quantum discord have a special status in that they are the most parsimonious of the measures in quantifying nonclassical correlations. WPM showed that their measure is nonnegative, and Datta [Dat08] showed that discord is nonnegative, allowing us to conclude that all the other measures are also nonnegative. Both proofs rely on the strong subadditivity of quantum entropy [NC00]; we review the proofs in App. A.2. Careful consideration of the conditions for saturating the strong-additivity inequality [HJPW04], not presented here, give the conditions for WPM and discord to be zero: the WPM measure is zero if and only if ρ AB is diagonal in a product basis, i.e., an orthonormal basis of the form e a f b, and discord is zero if and only if ρ AB is diagonal in a conditional product basis (pointing from A to B), i.e., an orthonormal basis of the form e a f b a. Since MID, like WPM, is zero if and only if ρ AB is diagonal in a product basis,

72 Chapter 4. A framework for entropic measures of nonclassical correlations 54 the relations in the array (4.51) imply that M 2b and M 3b are zero if and only if ρ AB is diagonal in a product basis. Similarly, the inequality M (dd) 3c M (discord) 2c shows that having ρ AB diagonal in a conditional product basis is necessary to make M 3c zero, and a moment s contemplation of Eqs. (4.48) (4.50) shows that this is also a sufficient condition. For pure states, we have S(A, B) = 0, S(A) = S(B) = S(B A) = S(A B), and S(A : B) = 2S(A) = 2S(B). It is easy to show that the optimal measurement for all the measures in the array is measurement in the Schmidt basis of the pure state (marginal eigenbasis for each subsystem), which gives H(A) = H(B) = H(A, B) = H(A : B) = S(A) = S(B) and H(B A) = H(A B) = 0. Thus all the measures in the array, except the quantum mutual information, are equal to the marginal quantum entropy, S(A) = S(B), which is the entropic measure of entanglement for bipartite pure states. The remaining gap in our understanding left by the relations in the array is whether there is an inequality between the WPM measure and discord. The WPM measure is strictly bigger than zero for states that are diagonal in a conditional product basis that is not a product basis and so is bigger than the quantum discord for such states. If there is an inequality, it must be that the WPM measure is bounded below by quantum discord. Indeed, it is not hard to come up with a proof, using the method of Piani et al. [PHH08]. The proof, given in App. A.2, is part of the two-step demonstration that WPM and discord are nonnegative. We conclude that M (WPM) 1b M (discord) 2c = D(A B). (4.53) The proof allows us to identify the equality condition: the WPM measure is equal to discord if and only if ρ AB is diagonal in a conditional product basis that points from B to A. We emphasize that product bases and conditional product bases do not exhaust the set of orthonormal bases that are made up of product states. There are orthonormal bases made up entirely of product states that are neither product bases nor conditional product bases; these have been studied, for example, in the context of nonlocality

73 Chapter 4. A framework for entropic measures of nonclassical correlations 55 without entanglement [BDF + 99]. Not surprisingly, we refer to such a basis as a basis of product states, to be distinguished from a product basis or a conditional product basis Rank-one POVMs and projective measurements In Sec we assumed that all the measurements were described by rank-one POVMs. This assumption does not affect the demon-based work deficit (4.42) for the case of erasure without communication, for that case, which leads to the quantum mutual information, does not rely on any assumptions about how the subsystems are measured. Nor does this assumption affect MID, which is derived from measurement strategy (a), a strategy that from the outset prescribes orthogonal-projection-valued measurements in the eigenbases of the marginal density operators. The assumption must be carefully examined, however, for measurement strategies (b) and (c). On the face of it, there is a problem for the second and third rows of our array. For type-2 measures, the task is to minimize a classical conditional entropy, and for type-3 measures, the task is to minimize a classical joint entropy. In both cases, the minimum is achieved by making no measurements at all. For the demon-based measures in the right two columns of the third row, it is clear what the problem is. The contribution of H(A, B) to the classical work comes from the erasure cost; the local demons can minimize their erasure cost by not having a measurement record. Of course, if the local demons make no measurements, they also cannot extract the work attendant on knowing more about their system s state. The upshot is that formula (4.44) for the net classical work needs to be modified if one does not assume measurements described by rank-one POVMs. App. A.4 shows, not surprisingly, that, once modified, the net classical work is always optimized on rank-one POVMs, so one can restrict the demons in this way without affecting their performance. For the measure M 2b, we know of no reason to restrict to rank-one POVMs more compelling than declaring that the measure would be nonsense without this restriction.

74 Chapter 4. A framework for entropic measures of nonclassical correlations 56 For the quantum discord, we can do better: the original definition of discord [OZ01] did not discuss conditioned measurements on B, but rather formulated the discord directly in terms of minimizing the classical conditional entropy as in Eqs. (4.37) (4.39); this is equivalent to our assuming rank-one POVMs for the measurement on B. We are still left with a question why should the measurements on A be restricted to rank-one POVMs? and this same question applies to both local measurements for the WPM measure. We now address this question by showing in both situations that the optimum can always be attained on rank-one POVMs. It is important to show this, because the proofs regarding nonnegativity and ordering of the WPM measure and discord, given in App. A.2, assume rank-one POVMs. We deal with the WPM measure first. The key point is obvious: making coarsegrained POVM measurements on A and B should not uncover as much mutual information as making fine-grained, rank-one POVM measurements. We start with POVMs {E a } and {F b } for systems A and B, and we imagine that these are a coarse graining of POVMs {E aj } and {F bk }, i.e., E a = j E aj, F b = k F bk. (4.54) A POVM element can always be fine-grained to the rank-one level by writing it in terms of its eigendecomposition. The joint probability for the fine-grained outcomes aj and bk is p ajbk = p jk ab p ab, with similar relations for the marginals for the two subsystems. It is now trivial to show that fine graining never decreases the classical mutual information: H(A, J : B, K) = H(A : B) + a,b p ab H(J : K a, b). (4.55) This means that in maximizing the classical mutual information, we need only consider rank-one POVMs. For the discord, the reduction to rank-one POVMs has been demonstrated by Datta [Dat08]; it is sufficiently brief that we repeat it here. Since the conditional measurements on B are already specified, we need only worry about fine graining the measurement on A. We need the conditional state of B given the coarse-grained

75 Chapter 4. A framework for entropic measures of nonclassical correlations 57 outcome a in terms of the conditional states given the fine-grained outcome aj: ρ B a = tr A(E a ρ AB ) p a = j tr A (E aj ρ AB ) p a = j p j a ρ B aj, (4.56) where p j a = p aj p a = tr A(E aj ρ A ) tr A (E a ρ A ). (4.57) The quantity to be minimized over measurements on A is the conditional entropy (4.37). For it, we can write H {Ea }(B A) = a = a p a S(ρ B a ) ( ) p a S p j a ρ B aj j (4.58) a,j p aj S(ρ B aj ) = H {Eaj }(B A), (4.59) where the inequality follows from the concavity of the von Neumann entropy. Thus fine graining never increases this conditional entropy, so we are assured that the minimum is attained on rank-one POVMs. We have now settled the question of restricting to rank-one POVMs for all the measures except the measure in the middle, M 2b, and for it, we simply assert that it makes sense only if we restrict to rank-one POVMs. A remaining question is whether we can further restrict to orthogonal-projection-valued measurements. Searching over the entire set of rank-one POVMs is a daunting task, considerably more onerous than searching just over projection-valued measurements. On this score, we can report that WPM drew attention to an example where the WPM measurements are optimized on a rank-one POVM that is not projection-valued; we extend this example to quantum discord and generalize it in App. A.3. These examples, however, require that at least one system have dimension bigger than two; For evaluating quantum discord for a two-qubit system, Chen et al. [CZY + 11] found, that there are some states for which three-element POVMs on system A do better than two-outcome, orthogonal-projection-valued measurements.

76 Chapter 4. A framework for entropic measures of nonclassical correlations 58 Exploring the situation numerically, Galve, Giorgi, and Zambrini [GGZ11] confirmed this finding, but suggested that the corrections to the two-qubit discord obtained by using POVMs, instead of orthogonal projectors, are negligible. In the next section, we do a wholesale evaluation of the various measures for two-qubit states; in the need for manageable numerics, we restrict the search over measurements to rank-one, orthogonal projection operators; according to [GGZ11], this should have no significant effect on the result. Several groups of investigators have considered Gaussian versions of nonclassicalcorrelation measures for Gaussian states of two harmonic-oscillator modes; the local measurements are restricted to Gaussian measurements, i.e., measurements whose POVM elements are the phase-space displacements of a particular single-mode Gaussian state. Giorda and Paris [GP10] and Addesso and Datta [AD10] focused on a Gaussian version of discord and showed that the optimal Gaussian measurements are rank-one POVMs, but that for some Gaussian states, the optimal measurement is not orthogonal-projection-valued and thus requires the use of POVMs. Mi sta et al. [MTG + 11] investigated Gaussian versions of MID and WPM (which they called AmeriolatedMID); their investigation showed that for some Gaussian states, the optimal Gaussian measurement for the WPM measure is not the globally optimal measurement when one allows nongaussian POVMs. 4.3 Numerical results for two-qubit states One purpose of our framework is to clarify relations among the various measures of nonclassical correlations beyond entanglement. The ordering of the measures is of particular interest. The framework provides by construction some ordering relations between the measures; in addition, we have proved, using the method of Piani et al., the important relation that the WPM measure is bounded below by the discord. Nonetheless, questions remain, in particular, of whether there is an ordering between M 2b and M (dd) 3c, as well as between M (WPM) 1b and M (dd) 3c.

77 Chapter 4. A framework for entropic measures of nonclassical correlations 59 Figure 4.1: M (discord) 2c = D(A B) plotted against M (WPM) 1b for one million randomly generated joint density matrices, using orthogonal projectors for the measurements. As expected, the WPM measure is never smaller than the discord; also evident is that discord is zero for a larger class of states than the WPM measure, those being the states that are diagonal in a conditional product basis pointing from A to B. In this section we illustrate and investigate the various orderings by presenting numerical evaluations of the several measures for randomly selected two-qubit states. It should be noted, however, that in order to do the optimizations over measurements Figure 4.2: M (dd) 3c plotted against M (WPM) 1b for 100,000 randomly generated joint density matrices. Since M (dd) 3c M (discord) 2c, the points from Fig. 4.1 move upwards. Many points pass the diagonal, and the ordering of Fig. 4.1 disappears.

78 Chapter 4. A framework for entropic measures of nonclassical correlations 60 Figure 4.3: M (dd) 3c plotted against M 2b for 100,000 randomly generated joint density matrices. Relative to Fig. 4.2, the points move right, due to the relation M 2b M 1b. Since not all of them pass the diagonal, there is no ordering relation between M 3c and M 2b. numerically, we have had to restrict ourselves to orthogonal projectors instead of the more general POVMs, so in some situations, we might not be finding the optimal measurements. To calculate the various correlation measures, we follow the approach of Al-Qasimi and James [AJ11]. The measurement operators E a and F b are orthogonal projectors, E a = e A a e A a, F b = e B b e B b, a, b {0, 1}, (4.60) e X 0 = cos θ X 0 + e iφx sin θ X 1, e X 1 = sin θ X 0 + e iφx cos θ X 1. (4.61) The required optimization is done by a numerical search over the angles {θ X, φ X } for X {A, B}. For measurement strategy (b), we must search over the four angles for both qubits, but for strategy (c), we need only search over the two angles for subsystem A. Figure 4.1 compares the WPM measure and discord, confirming the expectation that the WPM measure is never smaller than discord. Figures 4.2 and 4.3 display the aforementioned pairs of correlation measures where our framework does not imply an ordering relation; the numerical data show that there is no ordering for these pairs.

79 Chapter 4. A framework for entropic measures of nonclassical correlations 61 Figure 4.4: (A) Discord (blue circles) and the WPM measure (yellow crosses) for one million randomly chosen two-qubit states, plotted against entanglement of formation, E f. As the correlations increase, the spread between entanglement and WPM or discord decreases. (B) Two superimposed histograms showing the distribution of discord and the WPM measure for ranges of values of E f : left histogram shows discord (red) and WPM (yellow) for the states of (A) corresponding to 0.1 E f 0.2; right histogram shows discord (blue) and WPM (green) corresponding to 0.3 E f 0.4. Another relation we have explored numerically is the one between the correlation measures and entanglement. Figure 4.4 shows discord and the WPM measure plotted against entanglement of formation, reproducing the plot in [AJ11] for discord, but providing new data for the WPM measure. The entanglement of formation is calculated using Wootters s analytical expression [Woo98], E f (ρ) = h ( (1+ 1 C 2 (ρ))/2 ). Here h(x) is the binary entropy, h(x) = x log x (1 x) log(1 x), and C(ρ) is the concurrence, given by C(ρ) = max(0, λ 1 λ 2 λ 3 λ 4 ), where the λ j s are the eigenvalues in decreasing order of the operator ρ ρ ρ, with ρ = (σy σ y )ρ (σ y σ y ). To avoid the slow numerical optimization procedures, analytical expressions for the correlation measures would be desirable. Yet only for very restricted classes of joint states are such expressions available. Girolami et al. [GPA10] 1 suggested that there is an analytical expression for the WPM measure for general two-qubit states. To understand their claim, we write the joint two-qubit state in terms of Pauli 1 In the published version of their paper Girolami et al. restriced the validity of their analytical expression to the set of two-qubit X-states [GPA11].

80 Chapter 4. A framework for entropic measures of nonclassical correlations 62 Figure 4.5: Deviation of the numerically obtained, optimal measurement vectors from the maximal singular vectors of the correlation matrix for the WPM measure. The joint states are a mixture of a pure product state with marginal spin (Bloch) vectors a = (1, 0, 0) and b = (1/ 2, 1/2, 1/2) and a mixed Bell-diagonal (zero marginal spin vectors) state, with correlation matrix c = diag( 0.9, 0.8, 0.7). The mixing parameter is ɛ, with ɛ = 0 corresponding to the product state and ɛ = 1 to the Bell-diagonal state. The green curve shows the cosine of the angle between the maximal right singular vector and the measurement vector on system B. The red curve is the cosine of the angle between maximal left singular vector and the measurement vector on system A. operators: ρ AB = 1 4 ( I AB + a σ A I B + I A b σ B + 3 c jk σj A j,k=1 ) σk B. (4.62) The correlation matrix, c jk = tr(σj A σk B ρ AB ), is not symmetric, but c T c can be diagonalized as c T cm j = λ 2 jm j. The eigenvalues, λ 2 j, are the squares of the singular values of c, and the eigenvectors are the right singular vectors of c. The correlation matrix maps the right singular vectors to the left singular vectors, cm j = λ j n j, and the left singular vectors, n j, are the eigenvectors of cc T. The claim of Girolami et al. was that the maximal left and right singular vectors, i.e., those corresponding to the largest singular value of c, specify the optimal measurements for the WPM measure. We can confirm that the measurement vectors for generic (randomly generated) two-qubit states are close to the maximal singular vectors of c for all three correlation measures that are based on measurement strategy (b), but it can be shown analytically that in general the singular vectors

81 Chapter 4. A framework for entropic measures of nonclassical correlations 63 are not the optimal measurement vectors. Moreover, there are examples where the deviation becomes obvious in the numerics. Figure 4.5 shows an example where the angle between the measurement vectors and the maximal singular vectors is noticeable in the calculation of the WPM measure. Similar plots can be obtained for the measures M 2b and M 3b. In this chapter we considered several entropic measures of nonclassical correlations. We presented a framework from which these different measures emerge logically. This enabled us to investigate ordering relations between these measures, by putting them onto equal footing. While some of these ordering relations are a direct consequence of the different measurement strategies employed in the definition of the framework, others were either proven analytically or numerical evidence was presented demonstrating the nonexistence of an ordering between some measures. In the next chapter we investigate some properties of these measures further by using a pictorial approach for a restricted set of two qubit states.

82 64 Chapter 5 Quantum Discord and the Geometry of Bell-Diagonal States Maintenance of quantum coherence is clearly important for quantum-informationprocessing protocols. Noise and decoherence, by turning pure states into mixed states, generally destroy quantum coherence. Efficient representation of quantum information requires that a quantum-information-processing system be composed of parts [BKCD02]. For multi-partite systems, quantum coherence is related to nonclassical correlations between the parts. One can use decoherence mechanisms to explore the nooks and crannies of nonclassical-correlation measures. There is no sudden death of discord [FAC + 10], as is suggested by the absence of open sets of classical states, but the nonanalyticity of nonclassical measures points to the possibility of sudden changes in derivatives. Investigation of the behavior of nonclassical measures under decoherence has begun [Col10, MCSV09, MWF + 10, MPM10], with a focus on the action of decoherence within the class of two-qubit states that are diagonal in the Bell basis. This focus is motivated by the fact that entanglement measures and nonclassical-correlation measures can be calculated explicitly for the Bell-diagonal states, thus allowing one to determine how these measures change under decoherence.

83 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 65 The Bell-diagonal states are a three-parameter set, whose geometry, including the subsets of separable and classical subsets, can be depicted in three dimensions [HHHH09, HH96]. Level surfaces of entanglement and nonclassical measures can be plotted directly on this three-dimensional geometry. The result is a complete picture, for this simple case, of the structure of entanglement and nonclassicality. We suggest that it is more illuminating to use this picture to explain how measures of entanglement and nonclassicality change along the one-dimensional trajectories traced out by decohering states, rather than the other way around. Hence we review and expand the pictorial approach here. Figure 5.1: Geometry of Bell-diagonal states. The tetrahedron T is the set of valid Bell-diagonal states. The Bell states β ab sit at the four vertices, the extreme points of T. The green octahedron O, specified by c 1 + c 2 + c 3 1 (λ ab 1/2), is the set of separable Bell-diagonal states. There are four entangled regions outside O, one for each vertex of T, in each of which the biggest eigenvalue λ ab is the one associated with the Bell state at the vertex. Classical states, i.e., those diagonal in a product basis, lie on the Cartesian axes. In Sec. 2.3 we have seen that an arbitrary two-qubit state can be written as: ρ = 1 4 ( I 4 + a σ I 2 + I 2 b σ + i,j c i,j σ i σ j ). (5.1) Any two-qubit state, satisfying a = 0 = b, i.e., having maximally mixed marginal density operators ρ A = I/2 = ρ B, can be brought to Bell-diagonal form by using

84 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 66 local unitary operations on the two qubits to diagonalize the correlation matrix c i,j = σ A j σ B k [Luo08b]. Bell-diagonal states of two qubits, A and B, consequently have a density operators of the form ρ AB = 1 4 ( I + 3 c j σj A σj B j=1 ) = a,b λ ab β ab β ab. (5.2) The eigenstates are the four Bell states β ab ( 0, b + ( 1) a 1, 1 b )/ 2, with eigenvalues λ ab = 1 ) (1 + ( 1) a c 4 1 ( 1) a+b c 2 + ( 1) b c 3. (5.3) A Bell-diagonal state is specified by a 3-tuple (c 1, c 2, c 3 ). The density operator ρ AB must be a positive operator, i.e., λ ab 0; the resulting region of Bell-diagonal states is the state tetrahedron T in Fig Separable Bell-diagonal states are those with positive partial transpose [HHHH09]. Partial transposition changes the sign of c 2, so operators with positive partial transpose occupy the reflection of T through the plane c 2 = 0; the region of separable Bell-diagonal states is the intersection of the two tetrahedra, which is the octahedron O of Fig. 5.1 [HH96]. The entanglement of formation E [HHHH09, Woo98] is a monotonically increasing function of Wootters s concurrence C [Woo98], which for Bell-diagonal states, is given by C = max(0, 2λ max 1), where λ max = max λ ab. The concurrence and the entanglement of formation are convex functions on T. They are zero for the separable states in the octahedron O. In each of the four entangled regions outside O, C and E are constant on planes parallel to the bounding face of O and increase as one moves outward through these planes toward the Bell-state vertex. We will talk about discord when investigating correlation measures in this setting because it has been a focus of recent work on decoherence and nonclassical correlations [Col10, MCSV09, MWF + 10, MPM10]. As we will show, however, everything we will talk about directly applies to all of the correlation measures discussed in the previous chapter, since for the set of Bell-diagonal states, all of these measures are equal.

85 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 67 Discord is defined as the difference between the mutual information I and the accessible classical correlation C [HV01], D = I C = S(B A) S(B A). (5.4) The quantum mutual information I is regarded as quantifying the total correlations in the joint state ρ AB. The quantum mutual information of Bell-diagonal states, I = 2 S(ρ AB ) = a,b λ ab log 2 (4λ ab ), (5.5) is a convex function on T. It has smooth level surfaces that bulge outward toward the vertices of T (see Fig. 5.2). The next step is to quantify purely classical correlations C in terms of information from measurements. As we restricted ourselves to Bell-diagonal states, we can make use of an analytical expression for C, derived by Luo [Luo08b], and circumvent the cumbersome optimization procedure: C =1 H 2 ( 1 + c 2 ) = 1 + c log 2 2 (1 + c) + 1 c log 2 2 (1 c), (5.6) where H 2 (p) = p log 2 p (1 p) log 2 (1 p) is the binary entropy and c = max c j. Luo calculated C for Bell-diagonal states under the assumption that the measurement is described by one-dimensional orthogonal projection operators. We will relax this assumption and extend Luo s result to rank-one POVMs here. If the measurement on system A is to be described by arbitrary rank-one POVM elements E k = q k (I + n k σ), (5.7) we have p k = q k and the state ρ B k of system B after the measurement outcome k is ρ B k = (I + d k σ)/2, (5.8) with d kj = c j n kj. Now, we have S(ρ B k ) = H 2 [(1 + d k )/2 H 2 [(1 + c)/2], (5.9)

86 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 68 since d k c. This shows that S(B A) H 2 [(1+c)/2], with equality for measurement of orthogonal projectors along the direction of maximum c j. From Eqs. (5.8) and (5.9) we see that the optimal measurement on system B is always aligned with the measurement on system A and does not depend on the outcome of the first measurement. As both of the optimal measurements are specified by orthogonal projectors, and the marginal density operators for either subsystem of Bell-diagonal state are degenerate, the measurements are performed in the marginal eigenbasis of both parts. This causes the hierarchy of correlation measures in Eq. (4.51) to collapse and demonstrates, for the set of Bell diagonal states, the equality of all the correlation measures considered in the previous chapter. The accessible classical information C, a convex function on T, is constant on the surfaces of cubes (or the portion of such a cube in T ) centered at the origin in Fig. 5.2 this introduces nonanalyticity and C increases monotonically with the size of the cube. As we have shown that discord equals the symmetric correlation measures, due to the symmetry of the Bell-diagonal states, it vanishes if and only if ρ AB is diagonal in a product basis e A j fk B. In our three dimensional picture of the Bell-diagonal state space, these purely classical states fall on the Cartesian axes [DacVB10] (see Fig. 5.1). Figure 5.3 plots level surfaces of discord for Bell-diagonal states. From these plots, it is clear that discord is quite a different beast from entanglement of formation, quantum mutual information, and the measure of classical correlations. Whereas E, I, and C generally increase outward from the origin, D increases away from the Cartesian axes, capturing an entropic notion of distance from classical states [MPS + 10, DacVB10]. In particular, as one moves outward along one of the constant-discord tubes of Fig. 5.3, the classical correlations and the total correlations of the quantum mutual information increase, but their difference, the nonclassical correlations as measured by discord, remains constant. At the vertices of O, I = C = 1 and D = E = 0. At the Bell-state vertices of T, I = 2 and C = D = E = 1, this being the

87 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 69 maximum value of discord for two qubits. In addition, E, I, and C are all convex, whereas discord is neither concave nor convex, as is evident from the plots in Fig. 5.3: one can mix two positive-discord states to get a zero-discord classical state, and one can mix two zero-discord classical states on different axes to get a positive-discord state. This argument and its conclusion are not special to Bell-diagonal states: mixing two discordant states can lead to a state that is diagonal in a product basis, and mixing two states that are diagonal in incompatible product bases generally leads to a discordant state. Mazzola, Piilo, and Maniscalco [MPM10] recently investigated the dynamics of classical and nonclassical correlations, as measured by discord, for two qubits under decoherence processes that preserve Bell-diagonal states. In particular, they considered independent phase-flip channels for the two qubits. The phase flips are implemented mathematically by random applications of σ z operators to the qubits. This decoherence process leaves c 3 unchanged, but flips the signs of c 1 and c 2 randomly, leading to exponential decay of c 1 and c 2 at the same rate. Mazzola and collaborators found that for the initial conditions they considered, the entanglement of formation decays to zero in a finite time sudden death of entanglement [YE09] but that the discord remains constant for a finite time and then decays, reaching zero at infinite time. This situation is depicted in terms of the surfaces of constant discord in Figure 5.4. The decohering-state trajectory is a straight line that runs along a tube of constant discord, until it encounters an intersecting tube, after which the discord decreases to zero when the state becomes fully classical. This behavior is generic for flip channels and initial conditions on edges of the state tetrahedron. We focus here on the phase-flip channel with initial conditions in the (+,, +)-octant, but analogous considerations apply to the other flip channels (bit and bit-phase) and to initial conditions on the other edges of T. Consider then initial conditions anywhere along the edge of T in this octant: c 1 (0) = 1 and 0 c 2 (0) = c 3 (0) 1. The trajectory under phase flips is a straight line c 3 = c 3 (0) = c 2 /c 1. Along this straight line, the eigenvalues λ ab factor into products of probabilities, (1 ± c 1 )/2 and (1 ± c 3 )/2, thus making S(ρ AB ) the entropy of two

88 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 70 independent binary random variables with these probabilities. This yields a quantum mutual information I = 2 H 2 [(1 + c 3 )/2] H 2 [(1 + c 1 )/2]. Furthermore, along the trajectory c = max(c 1, c 3 ). The result is that the trajectory initially runs along a tube of constant discord D = 1 H 2 ( 1 + c3 2 ), (5.10) for c 1 c 3. When c 1 = c 3, the trajectory encounters another tube, after which, for c 1 c 3, the discord decreases monotonically as D = 1 H 2 [(1 + c 1 )/2] as c 1 decreases. Meanwhile, the entanglement of formation decreases monotonically from its initial value to a sudden death at c 1 = (1 c 3 )/(1 + c 3 ). The situation investigated in [MPM10] is surely interesting: under decoherence, nonclassical correlations remain constant for a finite time interval. This situation is, however, a special one, as can be seen from the surfaces of constant discord; the trajectories considered here are the only straight lines in parameter space that stay on a surface of constant discord. Indeed, the pictorial approach can provide a complete understanding of how entanglement and nonclassicality change under decoherence within the set of Bell-diagonal states.

89 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 71 (a) (c) (b) (d) Figure 5.2: Level surfaces of the quantum mutual information I (left column) and the accessible classical information C (right column): (a) I = 0.1; (b) I = 0.55; (c) C = 0.15; (d) C = 0.4. The smooth surface of cali = 0.1 bulges towards the vertices of the tetrahedron T from Fig. 5.1; as the mutual information grows, this surface becomes inflated and eventually intersects T, giving rise to the windows seen in (b) for I = The level surfaces for the accessible classical information C are cubes centered at the origin. As C increases corners of the cube get cut of as they poke through the surface of T.

90 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 72 (a) (b) (c) Figure 5.3: Surfaces of constant discord: (a) D = 0.03; (b) D = 0.15; (c) D = The level surfaces consist of three intersecting tubes running along the three Cartesian axes. The tubes are cut off by the state tetrahedron T at their ends, and they are squeezed and twisted so that at their ends, they align with an edge of T. As discord decreases, the tubes collapse to the Cartesian axes [DacVB10]. As discord increases, the tube structure is obscured, as in (c): the main body of each tube is cut off by T ; all that remains are the tips, which reach out toward the Bell-state vertices.

91 Chapter 5. Quantum Discord and the Geometry of Bell-Diagonal States 73 Figure 5.4: Trajectory (red) of a Bell-diagonal state under random phase flips of the two qubits; initial conditions are c 1 (0) = 1, c 2 (0) = c 3 (0) = 0.3. The trajectory is the straight line c 3 = c 3 (0) = 0.3 = c 2 /c 1. For clarity, only the (+,, +)-octant is shown. A constant-discord surface is plotted for the discord value of the initial state. Faces of the yellow state tetrahedron T and the green separable octahedron O are also shown. The straight-line trajectory proceeds along a tube of constant discord till it encounters the vertical tube at c 1 = 0.3, after which discord decreases monotonically to zero when the trajectory reaches the c 3 axis. Entanglement of formation decreases monotonically to zero when the trajectory enters O at c 1 = 0.7/1.3 = 0.54.